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Proposition 6.3.1 Let \( \mathcal{A} \) and \( \mathcal{B} \) be two monotone increasing families of subsets of \( N = \{ 1,2,\ldots, n\} \) and let \( \mathcal{C} \) and \( \mathcal{D} \) be two monotone decreasing families of subsets of \( N \) . Then\n\n\[ \Pr \left( {\mathcal{A} \cap \mathcal{B}}\right) \geq \Pr \l... | Proof. Let \( f : P\left( N\right) \rightarrow {\mathbb{R}}^{ + } \) be the characteristic function of \( \mathcal{A} \), i.e., \( f\left( A\right) = 0 \) if \( A \notin \mathcal{A} \) and \( f\left( A\right) = 1 \) if \( A \in \mathcal{A} \) . Similarly, let \( g \) be the characteristic function of \( B \) . By the a... | Yes |
\[ \alpha \left( G\right) \geq \mathop{\sum }\limits_{{v \in V}}\frac{1}{{d}_{v} + 1} \] | Proof. Let \( < \) be a uniformly chosen total ordering of \( V \) . Define\n\n\[ I = \{ v \in V : \{ v, w\} \in E \Rightarrow v < w\} \]\n\nLet \( {X}_{v} \) be the indicator random variable for \( v \in I \) and \( X = \mathop{\sum }\limits_{{v \in V}}{X}_{v} = \left| I\right| \) . For each \( v \)\n\n\[ E\left\lbrac... | Yes |
Theorem 2 [Turán (1941) ]Turán's Theorem Let \( H \) have \( n \) vertices and e edges. Then \( \alpha \left( H\right) \geq m \) and \( \alpha \left( H\right) = m \Leftrightarrow H \cong {G}_{n, e} \) . | Proof. \( {G}_{n, e} \) has \( \mathop{\sum }\limits_{{v \in V}}{\left( {d}_{v} + 1\right) }^{-1} = m \) since each clique contributes 1 to the sum. Fixing \( e = \mathop{\sum }\limits_{{v \in V}}{d}_{v}/2,\mathop{\sum }\limits_{{v \in V}}{\left( {d}_{v} + 1\right) }^{-1} \) is minimized with the \( {d}_{v} \) as close... | Yes |
Theorem 7.2.1 [Azuma’s Ineqality] Let \( 0 = {X}_{0},\ldots ,{X}_{m} \) be a martingale with\n\n\[ \left| {{X}_{i + 1} - {X}_{i}}\right| \leq 1 \]\n\nfor all \( 0 \leq i < m \) . Let \( \lambda > 0 \) be arbitrary. Then\n\n\[ \Pr \left\lbrack {{X}_{m} > \lambda \sqrt{m}}\right\rbrack < {e}^{-{\lambda }^{2}/2}. \] | Proof. Set, with foresight, \( \alpha = \lambda /\sqrt{m} \) . Set \( {Y}_{i} = {X}_{i} - {X}_{i - 1} \) so that \( \left| {Y}_{i}\right| \leq 1 \) and \( E\left\lbrack {{Y}_{i} \mid {X}_{i - 1},{X}_{i - 2},\ldots ,{X}_{0}}\right\rbrack = 0 \) . Then, as in A.1.16,\n\n\[ E\left\lbrack {{e}^{\alpha {Y}_{i}} \mid {X}_{i ... | Yes |
Theorem 7.2.3 When \( f \) satisfies the edge Lipschitz condition, the corresponding edge exposure martingale satisfies \( \left| {{X}_{i + 1} - {X}_{i}}\right| \leq 1 \) . When \( f \) satisfies the vertex Lipschitz condition the corresponding vertex exposure martingale satisfies \( \mid {X}_{i + 1} - \) \( {X}_{i} \m... | We prove these results in a more general context later. They have the intuitive sense that if knowledge of a particular vertex or edge cannot change \( f \) by more than one then exposing a vertex or edge should not change the expectation of \( f \) by more than one. | No |
Theorem 7.2.4 [Shamir and Spencer (1987) ] Let \( n, p \) be arbitrary and let \( c = \) \( E\left\lbrack {\chi \left( G\right) }\right\rbrack \) where \( G \sim G\left( {n, p}\right) \) . Then \[ \Pr \left\lbrack {\left| {\chi \left( G\right) - c}\right| > \lambda \sqrt{n - 1}}\right\rbrack < 2{e}^{-{\lambda }^{2}/2}.... | Proof. Consider the vertex exposure martingale \( {X}_{1},\ldots ,{X}_{n} \) on \( G\left( {n, p}\right) \) with \( f\left( G\right) = \) \( \chi \left( G\right) \) . A single vertex can always be given a new color so the vertex Lipschitz condition applies. Now apply Azuma’s Inequality in the form of Corollary 7.2.2. \... | Yes |
Lemma 7.3.1 \( E\left\lbrack Y\right\rbrack \geq \frac{{n}^{2}}{2{k}^{4}}\left( {1 + o\left( 1\right) }\right) \) . | Proof. Let \( \mathcal{K} \) denote the family of \( k \) -cliques of \( G \) so that \( f\left( k\right) = \mu = E\left\lbrack \left| \mathcal{K}\right| \right\rbrack \) . Let \( W \) denote the number of unordered pairs \( \{ A, B\} \) of \( k \) -cliques of \( G \) with \( 2 \leq \left| {A \cap B}\right| < k \) . Th... | Yes |
Theorem 7.3.2\n\n\[ \Pr \left\lbrack {\omega \left( G\right) < k}\right\rbrack < {e}^{-\left( {c + o\left( 1\right) }\right) \frac{{n}^{2}}{{\ln }^{8}n}} \] | Proof. Let \( {Y}_{0},\ldots ,{Y}_{m}, m = \left( \begin{array}{l} n \\ 2 \end{array}\right) \), be the edge exposure martingale on \( G\left( {n,1/2}\right) \) with the function \( Y \) just defined. The function \( Y \) satisfies the edge Lipschitz condition\nas adding a single edge can only add at most one clique to... | Yes |
Lemma 7.3.4 Let \( \alpha, c \) be fixed \( \alpha > \frac{5}{6} \) . Let \( p = {n}^{-\alpha } \) . Then almost always every \( c\sqrt{n} \) vertices of \( G = G\left( {n, p}\right) \) may be \( 3 \) -colored. | Proof. If not, let \( T \) be a minimal set which is not 3-colorable. As \( T - \{ x\} \) is 3-colorable, \( x \) must have internal degree at least 3 in \( T \) for all \( x \in T \) . Thus if \( T \) has \( t \) vertices it must have at least \( \frac{3t}{2} \) edges. The probability of this occuring for some \( T \)... | Yes |
Theorem 7.4.1 Let \( L \) satisfy the Lipschitz condition. Then the corresponding martingale satisfies\n\n\[ \left| {{X}_{i + 1}\left( h\right) - {X}_{i}\left( h\right) }\right| \leq 1 \]\n\nfor all \( 0 \leq i < m, h \in {A}^{B} \) . | Proof. Let \( H \) be the family of \( {h}^{\prime } \) which agree with \( h \) on \( {B}_{i + 1} \) . Then\n\n\[ {X}_{i + 1}\left( h\right) = \mathop{\sum }\limits_{{{h}^{\prime } \in H}}L\left( {h}^{\prime }\right) {w}_{{h}^{\prime }} \]\n\nwhere \( {w}_{{h}^{\prime }} \) is the conditional probability that \( g = {... | Yes |
For all \( \epsilon > 0 \) there exists \( \delta > 0 \) so that the following holds. Suppose Paul has a strategy for finding \( Y \) such that every line of questioning has total variance at most \( {\sigma }^{2} \) . Then \[ \Pr \left\lbrack {\left| {Y - E\left\lbrack Y\right\rbrack }\right| > {\alpha \sigma }}\right... | For simplicity we replace \( Y \) by \( Y - E\left\lbrack Y\right\rbrack \) so that we shall henceforth assume \( E\left\lbrack Y\right\rbrack = 0 \) . By symmetry we shall bound only the upper tail of \( Y \) . We set, with foresight, \( \lambda = \alpha /\left\lbrack {\sigma \left( {1 + \epsilon }\right) }\right\rbra... | Yes |
Theorem 7.5.2\n\n\[ \Pr \left\lbrack {X - E\left\lbrack X\right\rbrack > \lambda \sqrt{n}}\right\rbrack < {e}^{-{\lambda }^{2}/2}, \]\n\n\[ \Pr \left\lbrack {X - E\left\lbrack X\right\rbrack < - \lambda \sqrt{n}}\right\rbrack < {e}^{-{\lambda }^{2}/2}. \] | Proof. Consider \( \{ - 1, + 1{\} }^{n} \) as the underlying probability space with all \( \left( {{\epsilon }_{1},\ldots ,{\epsilon }_{n}}\right) \) equally likely. Then \( X \) is a random variable and we define a martingale \( {X}_{0},\ldots ,{X}_{n} = \) \( X \) by exposing one \( {\epsilon }_{i} \) at a time. The ... | Yes |
Theorem 7.5.3 Let \( \epsilon ,\lambda > 0 \) satisfy \( {e}^{-{\lambda }^{2}/2} = \epsilon \) . Then \[ \left| A\right| \geq \epsilon {2}^{n} \Rightarrow \left| {B\left( {A,{2\lambda }\sqrt{n}}\right) }\right| \geq \left( {1 - \epsilon }\right) {2}^{n}. \] | Proof. Consider \( \{ 0,1{\} }^{n} \) as the underlying probability space, all points equally likely. For \( y \in \{ 0,1{\} }^{n} \) set \[ X\left( y\right) = \mathop{\min }\limits_{{x \in A}}\rho \left( {x, y}\right) \] Let \( {X}_{0},{X}_{1},\ldots ,{X}_{n} = X \) be the martingale given by exposing one coordinate o... | Yes |
\[ \rho \left( {A,\overrightarrow{x}}\right) = \mathop{\min }\limits_{{\overrightarrow{v} \in V\left( {A,\overrightarrow{x}}\right) }}\left| \overrightarrow{v}\right| \] | Proof. Let \( \overrightarrow{v} \in V\left( {A,\overrightarrow{x}}\right) \) achieve this minimum. The hyperplane through \( \overrightarrow{v} \) perpendicular to the line from the origin to \( \overrightarrow{v} \) then separates \( V\left( {A,\overrightarrow{x}}\right) \) from the origin so that all \( \overrightar... | Yes |
Theorem 7.6.2\n\n\[ \int_{\Omega} \exp \left[ \frac{1}{4} \rho^2(A, \overrightarrow{x}) \right] d\overrightarrow{x} \leq \frac{1}{\Pr[A]}. \] | Proof.[Theorem 7.6.2] We use induction on the dimension \( n \). For \( n = 1, \rho(A, \overrightarrow{x}) = 1 \) if \( \overrightarrow{x} \notin A \), zero otherwise so that\n\n\[ \int \exp \left[ \frac{1}{4} \rho^2(A, \overrightarrow{x}) \right] = \Pr[A] + (1 - \Pr[A]) e^{1/4} \leq \frac{1}{\Pr[A]} \]\n\nas the inequ... | Yes |
Theorem 7.7.1 Under the above assumptions and for all \( b, t \)\n\n\[ \Pr \left\lbrack {X \leq b - t\sqrt{f\left( b\right) }}\right\rbrack \Pr \left\lbrack {X \geq b}\right\rbrack \leq {e}^{-{t}^{2}/4}. \] | Proof. Set \( A = \{ x : h\left( x\right) < b - t\sqrt{f\left( b\right) }\} \) . Now suppose \( h\left( y\right) \geq b \) . We claim \( y \notin {A}_{t} \) . Let \( I \) be a set of indices of size at most \( f\left( b\right) \) that certifies \( h\left( y\right) \geq b \) as given above. Define \( {\alpha }_{i} = 0 \... | Yes |
Theorem 1 [Weierstrass Approximation Theorem] For every continuous real function \( f : \left\lbrack {0,1}\right\rbrack \mapsto R \) and every \( \epsilon > 0 \), there is a polynomial \( p\left( x\right) \) such that \( \left| {p\left( x\right) - f\left( x\right) }\right| \leq \epsilon \) for all \( x \in \left\lbrack... | Proof. Since a continuous \( f : \left\lbrack {0,1}\right\rbrack \mapsto R \) is uniformly continuous there is a \( \delta > 0 \) such that if \( x,{x}^{\prime } \in \left\lbrack {0,1}\right\rbrack \) and \( \left| {x - {x}^{\prime }}\right| \leq \delta \) then \( \left| {f\left( x\right) - f\left( {x}^{\prime }\right)... | Yes |
Theorem 8.1.1 [The Janson Inequality] Let \( {B}_{i}, i \in I,\Delta, M,\mu \) be as above and assume all \( \Pr \left\lbrack {B}_{i}\right\rbrack \leq \epsilon \) . Then\n\n\[ M \leq \Pr \left\lbrack {{ \land }_{i \in I}\overline{{B}_{i}}}\right\rbrack \leq M{e}^{\frac{1}{1 - \epsilon }\frac{\Delta }{2}} \]\n\nand, fu... | For each \( i \in I \)\n\n\[ \Pr \left\lbrack \overline{{B}_{i}}\right\rbrack = 1 - \Pr \left\lbrack {B}_{i}\right\rbrack \leq {e}^{-\Pr \left\lbrack {B}_{i}\right\rbrack } \]\n\nso, multiplying over \( i \in I \) ,\n\n\[ M \leq {e}^{-\mu }. \] | No |
Theorem 8.3.1 Suppose there is a constant \( \mu \) so that\n\n\[ E\left\lbrack X\right\rbrack = {S}^{\left( 1\right) } \rightarrow \mu \]\n\nand such that for every fixed \( r \)\n\n\[ E\left\lbrack {{X}^{\left( r\right) }/r!}\right\rbrack = {S}^{\left( r\right) } \rightarrow {\mu }^{r}/r! \]\n\nThen\n\n\[ \Pr \left\l... | Proof. We do only the case \( t = 0 \) . Fix \( \epsilon > 0 \) . Choose \( s \) so that\n\n\[ \left| {\mathop{\sum }\limits_{{r = 0}}^{{2s}}{\left( -1\right) }^{r}\frac{{\mu }^{r}}{r!} - {e}^{-\mu }}\right| \leq \frac{\epsilon }{2}. \]\n\nThe Bonferroni Inequalities state that, in general, the inclusion-exclusion form... | Yes |
Lemma 8.4.1 With the above notation and for any integer \( s \)\n\n\[ \Pr \left\lbrack {\text{ there exists a disfam }J,\left| J\right| = s}\right\rbrack \leq \frac{{\mu }^{s}}{s!}. \] | Proof. Let \( \mathop{\sum }\limits^{ * } \) denote the sum over all \( s \) -sets \( J \subseteq I \) with no \( j \sim {j}^{\prime } \) . Let \( \mathop{\sum }\limits^{o} \) denote the sum over ordered \( s \) -tuples \( \left( {{j}_{1},\ldots ,{j}_{s}}\right) \) with \( \left\{ {{j}_{1},\ldots ,{j}_{s}}\right\} \) f... | Yes |
Lemma 8.4.2 With the above notation and for any integer \( s \)\n\n\[\n\begin{aligned} \Pr \left\lbrack {\text{ there exists a maxdisfam }J,\left| J\right| = s}\right\rbrack & \leq \frac{{\mu }^{s}}{s!}{e}^{-{\mu }_{s}}{e}^{\frac{\Delta }{2}} \\ & \leq \frac{{\mu }^{s}}{s!}{e}^{-\mu }{e}^{s\nu }{e}^{\frac{\Delta }{2}}.... | Proof. As in Lemma 8.4.1 we bound this probability by \( \mathop{\sum }\limits^{ * } \) of \( J = \left\{ {{j}_{1},\ldots ,{j}_{s}}\right\} \) being a maxdisfam. For this to occur \( J \) must first be a disfam and then \( { \land }^{ * }\overline{{B}_{i}} \), where \( { \land }^{ * } \) is the conjunction over all \( ... | Yes |
Theorem 8.5.1 Set \( p = \frac{\ln n}{n}\omega \left( n\right) \) where \( \omega \left( n\right) \rightarrow \infty \) arbitrarily slowly. Then in \( G\left( {n, p}\right) \) almost always | \[ \deg \left( x\right) \sim \left( {n - 1}\right) p \] for all vertices \( \mathbf{x} \). This is actually a large deviation result. It suffices to show the following. | Yes |
Theorem 8.5.2 Set \( p = \frac{\ln n}{n}\omega \left( n\right) \) where \( \omega \left( n\right) \rightarrow \infty \) arbitrarily slowly. Let \( x \in G \) be fixed. Fix \( \epsilon > 0 \) . Then\n\n\[ \Pr \left\lbrack {\left| {\deg \left( x\right) - \left( {n - 1}\right) p}\right| > \epsilon \left( {n - 1}\right) p}... | Proof. As \( \deg \left( x\right) \sim B\left( {n - 1, p}\right) \), i.e., it is a Binomial random variable with the above parameters, we have from A.1.14 that\n\n\[ \Pr \left\lbrack {\left| {\deg \left( x\right) - \left( {n - 1}\right) p}\right| > \epsilon \left( {n - 1}\right) p}\right\rbrack < 2{e}^{-{c}_{\epsilon }... | Yes |
Theorem 8.5.4 Let \( p \) be such that \( \mu > > \ln n \) . Let \( x \in G \) be fixed. Fix \( \epsilon > 0 \) Then\n\n\[ \Pr \left\lbrack {\left| {N\left( x\right) - \mu }\right| > {\epsilon \mu }}\right\rbrack = o\left( {n}^{-1}\right) . \] | Proof. We shall prove this under the further assumption \( p = {n}^{-2/3 + o\left( 1\right) } \) (or, equivalently, \( \mu = {n}^{o\left( 1\right) } \) ) which could be removed by technical methods. We now have, in the notation of Lemmas 8.4.1,8.4.2 \( {\nu \mu },\Delta = o\left( 1\right) \) . Let \( P \) denote the Po... | Yes |
Theorem 8.6.2 [Erdős (1956)] There is a set \( S \) for which \( f\left( n\right) = \Theta \left( {\ln n}\right) \) . That is, there is a set \( S \) and constants \( {c}_{1},{c}_{2} \) so that for all sufficiently large \( n \)\n\n\[ \n{c}_{1}\ln n \leq f\left( n\right) \leq {c}_{2}\ln n \n\] | Proof. Define \( S \) randomly by\n\n\[ \n\Pr \left\lbrack {x \in S}\right\rbrack = {p}_{x} = \min \left\lbrack {{10}\sqrt{\frac{\ln x}{x}},1}\right\rbrack . \n\]\n\nFix \( n \) . Now \( f\left( n\right) \) is a random variable with mean\n\n\[ \n\mu = E\left\lbrack {f\left( n\right) }\right\rbrack = \frac{1}{2}\mathop{... | Yes |
Theorem 8.6.3 [Erdős and Tetali (1990) ] There is a set \( S \) for which \( g\left( n\right) = \) \( \Theta \left( {\ln n}\right) \) . That is, there is a set \( S \) and constants \( {c}_{1},{c}_{2} \) so that for all sufficiently large \( n \)\n\n\[{c}_{1}\ln n \leq g\left( n\right) \leq {c}_{2}\ln n\] | Proof. Define \( S \) randomly by\n\n\[ \Pr \left\lbrack {x \in S}\right\rbrack = {p}_{x} = \min \left\lbrack {{10}{\left( \frac{\ln x}{{x}^{2}}\right) }^{1/3},\frac{1}{2}}\right\rbrack . \]\n\nFix \( n \) . Now \( g\left( n\right) \) is a random variable and\n\n\[ \mu = E\left\lbrack {g\left( n\right) }\right\rbrack =... | Yes |
For all \( k \) there exists \( \epsilon > 0 \) so that for all sufficiently large \( n \) there exist graphs \( G \) on \( n \) vertices with \( \chi \left( G\right) > k \) and yet \( \chi \left( {\left. G\right| }_{S}\right) \leq 3 \) for every set \( S \) of vertices of size at most \( {\epsilon n} \) . | Proof. For a given \( k \) let \( c,\epsilon > 0 \) satisfy (with foresight)\n\n\[ c > 2{k}^{2}H\left( {1/k}\right) \ln 2 \]\n\n\[ \epsilon < {e}^{-5}{3}^{3}{c}^{-3} \]\n\nwhere \( H\left( x\right) = - x{\log }_{2}x - \left( {1 - x}\right) {\log }_{2}\left( {1 - x}\right) \) is the entropy function. Set \( p = c/n \) a... | Yes |
Theorem 9.1.1 For the tournaments \( {T}_{p} \) described above,\n\n\[ c\left( {T}_{p}\right) \leq \frac{1}{2}\left( \begin{array}{l} p \\ 2 \end{array}\right) + O\left( {{p}^{3/2}\log p}\right) \] | In order to prove this theorem we need some preparations. Let \( \chi \) be the quadratic residue character defined on the elements of the finite field \( {GF}\left( p\right) \) by \( \chi \left( y\right) = \) \( {y}^{\left( {p - 1}\right) /2} \) . Equivalently, \( \chi \left( y\right) \) is 1 if \( y \) is a nonzero s... | No |
Lemma 9.1.2 For any two subsets \( A \) and \( B \) of \( {GF}\left( p\right) \); \( \left| {\mathop{\sum }\limits_{{i \in A}}\mathop{\sum }\limits_{{j \in B}}{d}_{ij}}\right| \leq {\left| A\right| }^{1/2}{\left| B\right| }^{1/2}{p}^{1/2}. \) | Proof. By the Cauchy-Schwarz Inequality and by the fact above: \[ {\left( \mathop{\sum }\limits_{{i \in A}}\mathop{\sum }\limits_{{j \in B}}{d}_{ij}\right) }^{2} \leq \left| A\right| \left( {\mathop{\sum }\limits_{{i \in A}}{\left( \mathop{\sum }\limits_{{j \in B}}{d}_{ij}\right) }^{2}}\right) \] \[ \leq \left| A\right... | Yes |
Theorem 9.2.1 For every partition of the set of vertices \( V \) into two disjoint subsets \( B \) and \( C \) :\n\n\[ e\left( {B, C}\right) \geq \frac{\left( {d - \lambda }\right) \left| B\right| \left| C\right| }{n}. \] | Proof. Put \( \left| V\right| = n, b = \left| B\right|, c = \left| C\right| = n - b \) . Let \( D = {dI} \) be the \( n \) by \( n \) scalar matrix with the degree of regularity of \( G \) on its diagonal. Observe that for any real vector \( x \) of length \( n \) (considered as a function \( x : V \mapsto R \) ) we ha... | Yes |
Corollary 9.2.2 If \( \lambda \) is the second largest eigenvalue of a d-regular graph \( G \) with \( n \) vertices, then \( G \) is an \( \left( {n, d, c}\right) \) -expander for \( c = \frac{d - \lambda }{2d} \) . | Proof. Let \( W \) be a set of \( w \leq n/2 \) vertices of \( G \) . By Theorem 9.2.1 there are at least \( \frac{\left( {d - \lambda }\right) w\left( {n - w}\right) }{n} \geq \frac{\left( {d - \bar{\lambda }}\right) w}{2} \) edges from \( W \) to its complement. Since no vertex in the complement is adjacent to more t... | Yes |
Theorem 9.2.4 Let \( G = \\left( {V, E}\\right) \) be a d-regular graph on \( n \) vertices, and suppose the absolute value of each of its eigenvalues but the first one is at most \( \\lambda \) . For a vertex \( v \\in V \) and a subset \( B \) of \( V \) denote by \( N\\left( v\\right) \) the set of all neighbours of... | Proof. Let \( A \) be the adjacency matrix of \( G \) and define a vector \( f : V \\mapsto R \) by \( f\\left( v\\right) = 1 - b \) for \( v \\in B \) and \( f\\left( v\\right) = - b \) for \( v \\notin B \) . Clearly \( \\mathop{\\sum }\\limits_{{v \\in V}}f\\left( v\\right) = 0 \), i.e., \( f \) is orthogonal to the... | Yes |
Corollary 9.2.5 Let \( G = \left( {V, E}\right), d, n \) and \( \lambda \) be as in Theorem 9.2.4. Then for every two sets of vertices \( B \) and \( C \) of \( G \), where \( \left| B\right| = {bn} \) and \( \left| C\right| = {cn} \) we have:\n\n\[ \left| {e\left( {B, C}\right) - {cbdn}}\right| \leq \lambda \sqrt{bc}n... | Proof. By Theorem 9.2.4\n\n\[ \mathop{\sum }\limits_{{v \in C}}{\left( \left| {N}_{B}\left( v\right) \right| - bd\right) }^{2} \leq \mathop{\sum }\limits_{{v \in V}}{\left( \left| {N}_{B}\left( v\right) \right| - bd\right) }^{2} \leq {\lambda }^{2}b\left( {1 - b}\right) n. \]\n\nThus, by the Cauchy Schwarz inequality;\... | Yes |
Theorem 9.2.7 Let \( G = \left( {V, E}\right) \) be a d-regular graph on \( n \) vertices, and suppose that each of its eigenvalues but the first one is at most \( \lambda \) . Let \( C \) be a set of \( {cn} \) vertices of \( G \) . Then, for every \( l \), the number of walks of length \( l \) in \( G \) that avoid \... | Proof. Let \( A \) be the adjacency matrix of \( G \) and let \( {A}^{\prime } \) be the adjacency matrix of its induced subgraph on the complement of \( C \) . We claim that the maximum eigenvalue of \( {A}^{\prime } \) is at most \( \left( {1 - c}\right) d + {c\lambda } \) . To prove this claim we must show that for ... | Yes |
Corollary 9.2.8 Let \( G = \left( {V, E}\right), d, n,\lambda, C \) and \( c \) be as in Theorem 9.2.7 and suppose\n\n\[ \left( {1 - c}\right) d + {c\lambda } \leq \frac{d}{\sqrt{2}} \]\n\nThen, for every \( l \), the probability that a randomly chosen walk of length \( l \) in \( G \) avoids \( C \) is at most \( {2}^... | Proof. The number of walks of length \( l \) in \( G \) that avoid \( C \) is at most \( \left( {1 - c}\right) n((1 - \) \( c)d + {c\lambda }{)}^{l} \leq n{d}^{l}{2}^{-l/2} \), by Theorem 9.2.7. Since the total number of walks is \( n{d}^{l} \) , the desired result follows. - | Yes |
Theorem 1 Let \( G = \left( {V, E}\right) \) be a vertex-transitive graph. For an integer \( k \) and for two (not neccessarily distinct) vertices \( u, v \) of \( G \), let \( {P}^{k}\left( {u, v}\right) \) denote the probability that a random walk of length \( k \) starting at \( u \) ends at \( v \) . Then, for ever... | Proof. We need the following simple inequality, sometimes attributed to Chebyschev.\n\nClaim 2 For every sequence \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) of \( n \) reals and for any permutation \( \pi \) of \( \{ 1,\ldots, n\} \) :\n\n\[ \n\mathop{\sum }\limits_{{i = 1}}^{n}{a}_{i}{a}_{\pi \left( i\right) } \leq... | Yes |
Theorem 10.1.1 Let \( H \) be a strictly balanced graph with \( v \) vertices, e edges and a automorphisms. Let \( c > 0 \) be arbitrary. Let \( A \) be the property that \( G \) contains no copy of \( H \) . Then with \( p = c{n}^{-v/e} \) , | Proof. Let \( {A}_{\alpha },1 \leq \alpha \leq \left( \begin{array}{l} n \\ v \end{array}\right) v!/a \), range over the edge sets of possible copies of \( H \) and let \( {B}_{\alpha } \) be the event \( G\left( {n, p}\right) \supseteq {A}_{\alpha } \) . We apply Janson’s Inequality. As\n\n\[ \mathop{\lim }\limits_{{n... | Yes |
Theorem 10.3.1 [Bollobás (1988) ] Almost always\n\n\\[ \chi \\left( G\\right) ) \\sim \\frac{n}{2{\\log }_{2}n}. \\] | Proof. Let \\( \\alpha \\left( G\\right) = \\omega \\left( \\bar{G}\\right) \\) denote, as usual, the independence number of \\( G \\) . The complement of \\( G \\) has the same distribution \\( G\\left( {n,1/2}\\right) \\) . Hence \\( \\alpha \\left( G\\right) \\leq \\left( {2 + o\\left( 1\\right) }\\right) {\\log }_{... | Yes |
Theorem 10.7.2 For any irrational \( \alpha ,0 < \alpha < 1 \), setting \( p = p\left( n\right) = {n}^{-\alpha } \), and for any First Order A, | Both proofs are only outlined. | No |
Lemma 10.7.3 For every First Order \( A \) there is a \( t = t\left( A\right) \) so that if \( G, H \) are any graphs with \( G \vDash A \) and \( H \vDash \neg A \) then Spoiler wins \( {EHR}\left\lbrack {G, H, t}\right\rbrack \) . | A detailed proof would require a formal analysis of the First Order language so we give only an example. Let \( A \) be the property \( {\forall }_{x}{\exists }_{y}\left\lbrack {x \sim y}\right\rbrack \) of not containing an isolated point and set \( t = 2 \) . Spoiler begins by selecting an isolated point \( {y}_{1} \... | No |
Theorem 10.7.4 A function \( p = p\left( n\right) \) satisfies the Zero-One Law if and only if for every \( t \), letting \( G\left( {n, p\left( n\right) }\right), H\left( {m, p\left( m\right) }\right) \) be independently chosen random graphs on disjoint vertex sets\n\n\[ \mathop{\lim }\limits_{{m, n \rightarrow \infty... | Proof. We prove only the \ | No |
Theorem 10.7.5 For any fixed \( p,0 < p < 1 \), and any \( s, G\left( {n, p}\right) \) almost always has the full level \( s \) extension property. | Proof. For every distinct \( {u}_{1},\ldots ,{u}_{a},{v}_{1},\ldots ,{v}_{b}, x \in G \) with \( a + b \leq s \) we define \( {E}_{{u}_{1},\ldots ,{u}_{a},{v}_{1},\ldots ,{v}_{b}, x} \) to be the event that \( \left\{ {x,{u}_{i}}\right\} \in E\left( G\right) ,1 \leq i \leq a \) and \( \left\{ {x,{v}_{j}}\right\} \notin... | Yes |
Lemma 10.7.6 [Generic Extension] Let \( \left( {R, H}\right) \), as given above, be safe. Let \( t \geq 0 \) be an arbitrary, but fixed, integer. Then in \( G \sim G\left( {n,{n}^{-\alpha }}\right) \) almost surely for all \( {x}_{1},\ldots ,{x}_{r} \) there exist \( {y}_{1},\ldots ,{y}_{v} \) such that\n\n1. \( {y}_{1... | Proof. From Exercise 5 almost surely every \( {x}_{1},\ldots ,{x}_{r} \) has \( \Theta \left( {{n}^{v}{p}^{e}}\right) \left( {R, H}\right) \) extensions \( {y}_{1},\ldots ,{y}_{v} \) . Our rough notion will be that the number of these \( {y}_{1},\ldots ,{y}_{v} \) that fail to be generic, in any of the bounded number o... | No |
Lemma 10.7.7 [Finite Closure] Let \( \alpha, r > 0 \) be fixed. Set \( \varepsilon \) equal to the minimal value of \( \frac{{e\alpha } - v}{v} \) over all integers \( v, e \) with \( 1 \leq v \leq t \) and \( {e\alpha } - v > 0 \) . Let \( K \) be such that \( r - {K\varepsilon } < 0 \) . Then in \( G\left( {n,{n}^{-\... | Proof. If not there would be a rigid \( t \) -chain \( X = {X}_{0} \subset {X}_{1} \subset \ldots \subset {X}_{L} = Y \) with \( K + r < \left| Y\right| < K + r + t \) . Letting \( \left( {{X}_{i - 1},{X}_{i}}\right) \) have type \( \left( {{v}_{i},{e}_{i}}\right) \) the restriction of \( G \) to \( Y \) would have \( ... | Yes |
Theorem 11.2.2 Let \( f = f\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) be a function and let \( C \) be a \( C\left( {\infty, s, d, t}\right) \) - circuit computing \( f \), where \( s \cdot {\left( \frac{1}{2}\right) }^{t} \leq {0.5} \) . Then either \( f \) or its complement \( \bar{f} \) has a minterm of size at most... | Proof. Let us apply to \( C \), repeatedly, \( d - 2 \) times a random \( 1/\left( {10t}\right) \) - restriction. Each of these random restrictions, when applied to any bottom subcircuit of depth 2, transforms it by Lemma 11.2.1 with probability at least \( 1 - {\left( \frac{1}{2}\right) }^{t} \) from a \( t \) -Or-And... | Yes |
Corollary 11.2.3 For any \( d \geq 2 \), there is no \( C\left( {\infty ,\frac{1}{2} \cdot {2}^{\frac{1}{10}{n}^{1/\left( {d - 1}\right) }}, d,\frac{1}{10}{n}^{1/\left( {d - 1}\right) }}\right) \) - circuit that computes the parity function \( f\left( {{x}_{1},\ldots ,{x}_{n}}\right) = {x}_{1} \oplus \cdots \oplus {x}_... | Proof. Assuming there is such a circuit we obtain, by Theorem 11.2.2, that the value of \( f \) can be fixed by assigning values to at most \( n - \frac{1}{2}{n}^{1/\left( {d - 1}\right) } + \frac{1}{10}{n}^{1/\left( {d - 1}\right) } < n \) variables. This is false, and hence there is no such circuit. | Yes |
Lemma 11.3.2 There is no polynomial \( P\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) over \( {GF}\left( 3\right) \) of degree at most \( \sqrt{n} \) which is equal to the parity of \( {x}_{1},\ldots ,{x}_{n} \) for a set \( S \) of at least \( {0.9} \cdot {2}^{n} \) distinct binary vectors \( \left( {{x}_{1},\ldots ,{x}_... | Proof. Suppose this is false, and suppose \( S \subset \{ 0,1{\} }^{n},\;\left| S\right| \geq {0.9} \cdot {2}^{n} \) and \( P\left( {{x}_{1},\ldots ,{x}_{n}}\right) = {x}_{1} \oplus \cdots \oplus {x}_{n} \) for all \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \in S \) . Define a polynomial \( Q = Q\left( {{y}_{1},\ldots ... | Yes |
Corollary 11.3.3 There is no circuit of depth \( d \) and size \( s \leq \frac{1}{10}{2}^{\frac{1}{2}{n}^{1/{2d}}} \) computing the parity of \( {x}_{1},{x}_{2},\ldots ,{x}_{n} \) using Not, And, Or and \( {\operatorname{Mod}}_{3} \) gates. | Proof. Suppose this is false and let \( C \) be such a circuit. Put \( \ell = \frac{1}{2} \cdot {n}^{1/{2d}} \) . By Lemma 11.3.1 there is a polynomial \( P = P\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) over \( {GF}\left( 3\right) \), whose degree is at most \( {\left( 2\ell \right) }^{d} = \sqrt{n} \), which is equal ... | Yes |
Lemma 11.4.2 For all members \( {M}_{i} \) of \( M\left( C\right) \)\n\n\[ \n{A}_{i} - \left( {\mathop{\bigcup }\limits_{{j \leq i}}{\delta }_{ \sqcap }^{j}}\right) \subseteq {M}_{i} \subseteq {A}_{i} \cup \underset{j \leq i}{ \cup }{\delta }_{ \sqcup }^{j}.\n\]\n\n(11.10) | Proof. We apply induction on \( i \) . For \( i < 0\;{M}_{i} = {A}_{i} \) and thus (11.10) holds. Assuming (11.10) holds for all \( {M}_{j} \) with \( j < i \) we prove it for \( i \) . If \( {A}_{i} = {A}_{\ell } \cup {A}_{k} \) , then, by the induction hypothesis\n\n\[ \n{M}_{i} = {M}_{\ell } \cup {M}_{k} \cup {\delt... | Yes |
Lemma 11.5.1 Let \( f = f\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) be a non-atom Boolean function of \( n \) variables. Then there is an \( i,1 \leq i \leq n \) and an \( \varepsilon \in \{ 0,1\} \) such that for the function \( g = f\left( {{x}_{1},\ldots ,{x}_{i - 1},\varepsilon ,{x}_{i + 1},\ldots ,{x}_{n}}\right) ... | Proof. Fix a formula \( F \) computing \( f \) with \( l = L\left( f\right) \) And and \( {Or} \) gates. \( F \) can be represented by a binary tree each of whose \( l + 1 \) leaves is labeled by an atom \( {x}_{i} \) or \( {\bar{x}}_{i} \) . Let us choose, randomly, a variable \( {x}_{i},1 \leq i \leq n \) according t... | Yes |
Corollary 11.5.2 If \( f = f\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) and \( L\left( f\right) \leq {\left( \frac{n}{k}\right) }^{3/2} - 1 \), then one can assign values to \( n - k \) variables so that the resulting function \( g \) is an atom. | Proof. Repeated application of Lemma 11.5.1 \( n - k \) times yields a \( g \) with\n\n\[ \left( {L\left( g\right) + 1}\right) \leq \mathop{\prod }\limits_{{i = k + 1}}^{n}{\left( 1 - \frac{1}{i}\right) }^{3/2}\left( {L\left( f\right) + 1}\right) = {\left( k/n\right) }^{3/2}\left( {L\left( f\right) + 1}\right) \leq 1.\... | Yes |
Theorem 1 Let \( \mathcal{F} \) be an antichain. Then\n\n\[ \mathop{\sum }\limits_{{A \in \mathcal{F}}}\frac{1}{\left( \begin{matrix} n \\ \left| A\right| \end{matrix}\right) } \leq 1 \] | Proof. Let \( \sigma \) be a uniformly chosen permutation of \( \{ 1,\ldots, n\} \) and set\n\n\[ {\mathcal{C}}_{\sigma } = \{ \{ \sigma \left( j\right) : 1 \leq j \leq i\} : 0 \leq i \leq n\} \]\n\n(The cases \( i = 0, n \) give \( \varnothing ,\{ 1,\ldots, n\} \in \mathcal{C} \) respectively.) Define a random variabl... | Yes |
Corollary 2 [Sperner’s Theorem] Let \( \mathcal{F} \) be an antichain. Then\n\n\[ \left| \mathcal{F}\right| \leq \left( \begin{matrix} n \\ \lfloor n/2\rfloor \end{matrix}\right) \] | Proof. The function \( \left( \begin{array}{l} n \\ x \end{array}\right) \) is maximized at \( x = \lfloor n/2\rfloor \) so that\n\n\[ 1 \geq \mathop{\sum }\limits_{{A \in \mathcal{F}}}\frac{1}{\left( \begin{matrix} n \\ \left| A\right| \end{matrix}\right) } \geq \frac{\left| \mathcal{F}\right| }{\left( \begin{matrix} ... | Yes |
Theorem 12.1.1 Let \( \mathcal{A} \) be a family of \( n \) subsets of an \( m \) -set \( \Omega \) . Then\n\n\[ \operatorname{disc}\left( \mathcal{A}\right) \leq \sqrt{{2m}\ln \left( {2n}\right) } \] | Proof. Let \( \chi : \Omega \rightarrow \{ - 1, + 1\} \) be random. For \( A \subset \Omega \) let \( {X}_{A} \) be the indicator random variable for \( \left| {\chi \left( A\right) }\right| > \alpha \) where we set \( \alpha = \sqrt{{2m}\ln \left( {2n}\right) } \) . If \( \left| A\right| = a \) then \( \chi \left( A\r... | Yes |
Theorem 12.2.1 [Spencer (1985a) ] Let \( \mathcal{A} \) be a family of \( n \) subsets of an \( n \) -element set \( \Omega \) . Then \[ \operatorname{disc}\left( \mathcal{A}\right) < 6\sqrt{n} \] | With \( \chi : \Omega \rightarrow \{ - 1, + 1\} \) random, \( A \in \mathcal{A},\chi \left( A\right) \) has zero mean and variance at most \( \sqrt{n} \) . If \( \left| {\chi \left( A\right) }\right| > 6\sqrt{n} \) then \( \chi \left( A\right) \) is at least six standard deviations off the mean. The probability of this... | No |
Lemma 12.2.3 Let \( \mathcal{A} \) be a family of \( n \) subsets of an \( r \) -set \( \Omega \) with \( r \leq {10}^{-9}n \) . Then there is a partial coloring \( \chi \) of \( \Omega \) with at most \( {10}^{-{40}}r \) points uncolored so that | We outline the argument which leaves room to spare. Let \( {A}_{1},\ldots ,{A}_{n} \) denote the sets of \( \mathcal{A} \) . Let \( \chi : \Omega \rightarrow \{ - 1, + 1\} \) be random. For \( 1 \leq i \leq n \) define\n\n\[ \n{b}_{i} = \text{ nearest integer to }\frac{\chi \left( {A}_{i}\right) }{{20}\sqrt{r}\sqrt{\ln... | No |
Theorem 12.3.1 Let \( \mathcal{A} \) be a family of \( n \) sets on \( m \) points with \( m \geq n \) . Suppose that lindisc \( \left( {\left. \mathcal{A}\right| }_{X}\right) \leq K \) for every subset \( X \) of at most \( n \) points. Then \( \overline{\operatorname{lindisc}}\left( \mathcal{A}\right) \leq K \) . | Proof. Let \( {p}_{1},\ldots ,{p}_{m} \in \left\lbrack {0,1}\right\rbrack \) be given. We define a reduction process. Call index \( j \) fixed if \( {p}_{j} \in \{ 0,1\} \), otherwise call it floating, and let \( F \) denote the set of floating indices. If \( \left| F\right| \leq n \) then halt. Otherwise, let \( {y}_{... | Yes |
Corollary 12.3.3 Let \( \mathcal{A} \) be a family of \( n \) sets on \( m \) points. Suppose disc \( \left( {\left. \mathcal{A}\right| }_{X}\right) \leq K \) for every subset \( X \) with at most \( n \) points. Then \( \operatorname{disc}\left( \mathcal{A}\right) \leq {2K} \) . | Proof. For every \( X \subseteq \Omega \) with \( \left| X\right| \leq n \), herdisc \( \left( {\left. \mathcal{A}\right| }_{X}\right) \leq K \) so by Theorem 12.3.2 lindisc \( \left( {\left. \mathcal{A}\right| }_{X}\right) \leq K \) . By Theorem 12.3.1 lindisc \( \left( \mathcal{A}\right) \leq \bar{K} \) . But\n\n\[ \... | Yes |
For any family \( \mathcal{A} \) of \( n \) sets of arbitrary size | Apply Theorem 12.2.1 and Corollary 12.3.3. | No |
Theorem 12.5.1 Let \( \mathcal{A} \) be a finite family of finite sets, no restriction on either the number of sets nor on the cardinality of the sets, with \( \deg \left( \mathcal{A}\right) \leq t \) . Then\n\n\[ \operatorname{disc}\left( \mathcal{A}\right) \leq {2t} - 1 \] | Proof. For convenience write \( \mathcal{A} = \left\{ {{A}_{1},\ldots ,{A}_{m}}\right\} \) with all \( {A}_{i} \subseteq \Omega = \{ 1,\ldots, n\} \) . To each \( j \in \Omega \) there is assigned a value \( {\mathbf{x}}_{\mathbf{j}} \) which will change as the proof progresses. Initially all \( {x}_{j} = 0 \) . At the... | No |
Theorem 13.1.1 For every \( d \geq 1 \) there is a set of at least \( \left\lfloor {\frac{1}{2}{\left( \frac{2}{\sqrt{3}}\right) }^{d}}\right\rfloor \) points in the \( d \) -dimensional Euclidean space \( {R}^{d} \), such that all angles determined by three points from the set are strictly less than \( \pi /2 \) . | Proof.[Theorem 13.1.1] We select the points of a set \( X \) in \( {R}^{d} \) from the vertices of the \( d \) -dimensional cube. As usual, we view the vertices of the cube, which are \( 0,1 \) -vectors of length \( d \), as the characteristic vectors of subsets of a \( d \) -element set; i.e., each 0,1-vector \( a \) ... | Yes |
Theorem 13.2.1 Let \( {I}_{1},{I}_{2},\ldots ,{I}_{n} \) be parallel unit intervals in the plane, where\n\n\[ \n{I}_{i} = \{ \left( {x, y}\right) : x = i,0 \leq y \leq 1\} .\n\]\n\nFor each \( i \) let us choose a point \( {p}_{i} \) randomly and independently from \( {I}_{i} \) according to a uniform distribution. Let... | Proof. We first estimate the probability that the triangle determined by the points \( {p}_{i},{p}_{i + a} \) and \( {p}_{i + k} \) is empty, for some fixed \( i, a \) and \( k = a + b \geq 3 \) . Let \( A = \left( {i, x}\right) \) , \( B = \left( {i + a, y}\right) \) and \( C = \left( {i + k, z}\right) \) be the point... | Yes |
Theorem 13.3.1 For all \( m \geq n \) ,\n\n\[ d\left( {m, n}\right) \leq \left( {n + 1}\right) /2 + \sqrt{\frac{n - 1}{2}\log m} \] | For the proof, we need a definition and two lemmas. For a vector \( \mathbf{a} = \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) of +1,-1-entries, the number of sign-changes in \( \mathbf{a} \) is the number of indices \( i \) , \( 1 \leq i \leq n - 1 \) such that \( {a}_{i} = - {a}_{i + 1} \) . For a matrix \( A \) of \( +... | No |
Lemma 13.3.2 For any matrix \( A \) of \( + 1, - 1 \) -entries, \( d\left( A\right) \leq s\left( A\right) + 1 \) . | Proof. Let \( A = \left( {a}_{i, j}\right) \) be an \( m \) by \( n \) matrix of \( + 1, - 1 \) entries and suppose \( s = s\left( A\right) \) . Let \( {t}_{1} < {t}_{2} < \ldots < {t}_{n} \) be arbitrary reals, and define \( n \) points \( {P}_{1},{P}_{2},\ldots ,{P}_{n} \) in \( {R}^{s + 1} \) by: \( {P}_{j} = \left(... | Yes |
Lemma 13.3.3 For every \( m \) by \( n \) matrix \( A \) of \( + 1, - 1 \) -entries there is a matrix \( B \) obtained from \( A \) by multiplying some of the columns of \( A \) by -1, such that \( s\left( B\right) \leq \left( {n - 1}\right) /2 + \sqrt{\frac{n - 1}{2}\log m} \) . | Proof. For each column of \( A \), randomly and independently, choose a number \( \epsilon \in \{ + 1, - 1\} \), where each of the two choices is equally likely, and multiply this column by \( \epsilon \) . Let \( B \) be the random sign-matrix obtained in this way. Consider an arbitrary fixed row of \( B \) . One can ... | Yes |
Lemma 13.4.1 If \( \\left( {X, R}\\right) \) is a range space of VC-dimension \( d \) with \( \\left| X\\right| = n \) points then \( \\left| R\\right| \\leq g\\left( {d, n}\\right) \) . | Proof. We apply induction on \( n + d \) . The assertion is trivially true for \( d = 0 \) and \( n = 0 \) . Assuming it holds for \( n \) and \( d - 1 \) and for \( n - 1 \) and \( d - 1 \) we prove it for \( n \) and \( d \) . Let \( S = \\left( {X, R}\\right) \) be a range space of VC-dimension \( d \) on \( n \) po... | Yes |
Corollary 13.4.3 Let \( \left( {X, R}\right) \) be a range space of VC-dimension \( d \geq 2 \), and let \( \left( {X,{R}_{h}}\right) \) be the range space on \( X \) in which \( {R}_{h} = \left\{ {\left( {{r}_{1} \cap \ldots \cap {r}_{h}}\right) : {r}_{1},\ldots {r}_{h} \in R}\right\} \) . Then \( {VC}\left( {X,{R}_{h... | Proof. Let \( A \) be an arbitrary subset of cardinality \( n \) of \( X \) . By Corollary 13.4.2 \( \left| {{P}_{R}\left( A\right) }\right| \leq g\left( {d, n}\right) \leq {n}^{d} \) . Since each member of \( {P}_{{R}_{h}}\left( A\right) \) is an intersection of \( h \) members of \( {P}_{R}\left( A\right) \) it follo... | Yes |
Theorem 13.4.4 There is a positive constant \( c \) such that if \( \left( {X, R}\right) \) is any range-space of VC-dimension at most \( d, A \subset X \) is a finite subset and \( \epsilon ,\delta > 0 \), then a random subset \( B \) of cardinality \( s \) of \( A \) where \( s \) is at least the minimum between \( \... | \[ \frac{c}{{\epsilon }^{2}}\left( {\operatorname{dlog}\frac{d}{\epsilon } + \log \frac{1}{\delta }}\right) \] is an \( \epsilon \) -sample for \( A \) with probability at least \( 1 - \delta \) . | Yes |
Theorem 13.4.5 Let \( \\left( {X, R}\\right) \) be a range space of VC-dimension \( d \), let \( A \) be a finite subset of \( X \) and supppose \( 0 < \\epsilon ,\\delta < 1 \) . Let \( N \) be a set obtained by \( m \) random independent draws from \( A \), where\n\n\[ m \\geq {max}\\left( {\\frac{4}{\\epsilon }{log}... | Proof.[Theorem 13.4.5] Let \( \\left( {X, R}\\right) \) be a range space with VC-dimension \( d \), and let \( A \) be a subset of \( X \) of cardinality \( \\left| A\\right| = n \) . Suppose \( m \) satisfies (13.2), and let \( N = \\left( {{x}_{1},\\ldots ,{x}_{m}}\\right) \) be obtained by \( m \) independent random... | Yes |
Lemma 13.5.2 Let \( \\left( {A,\\mathcal{F}}\\right), n, h, t \) and \( c \) be as above, let \( B \) be a finite subset of \( p > 1 \) points of \( A \), and let \( \\mathcal{G} \) be a collection of \( m \) (not necessarily distinct) members of \( \\mathcal{F} \). Then there are two distinct points \( \\mathbf{x}, y ... | Proof. We may and will assume that \( p \) is larger than \( c + 1 \). Let \( g \) be the largest integer such that \( c{g}^{t} \\leq p - 1 \), that is, \( g = \\left\\lfloor {\\left( \\frac{p - 1}{c}\\right) }^{1/t}\\right\\rfloor \). Let \( L \) be a random collection of \( g \) members of \( \\mathcal{G} \), each pi... | Yes |
Theorem 13.5.3 Let \( \left( {A,\mathcal{F}}\right) \) be a finite range space, where \( \left| A\right| = n \), and suppose that its dual shatter function \( h \) satisfies \( h\left( g\right) \leq c{g}^{t} \) for some fixed \( c, t > 0 \) . Then, there is a \( {C}^{\prime } = {C}^{\prime }\left( {c, t}\right) \) such... | Proof. Without loss of generality, assume that the number of points of \( A \) is even (otherwise, simply omit a point). By Theorem 13.5.1 there is a Hamilton path \( {x}_{1}{x}_{2}\ldots {x}_{n} \) on these points such that each member of \( \mathcal{F} \) is stabbed by at most \( C{n}^{1 - 1/t}\log n \) edges of the ... | Yes |
Theorem 1 Let \( C \) be bounded, convex and centrally symmetric about the origin. Then\n\n\[ \delta \left( C\right) \geq {2}^{-\left( {n - 1}\right) } \] | Proof. Fix \( \epsilon > 0 \) . Normalize so \( \mu = \mu \left( C\right) = 2 - \epsilon \) . For any real \( z \) let \( {C}_{z} \) denote the \ | No |
Theorem 14.1.1 [Shannon’s Theorem] Let \( p \in \left( {0,{.5}}\right) \) be fixed. For \( \epsilon > 0 \) arbitrarily small there exists a Coding Scheme with rate of transmission greater than \( 1 - H\left( p\right) - \epsilon \) and probability of incorrect transmission less than \( \epsilon \) . | Proof. Let \( \delta > 0 \) be such that \( p + \delta < {.5} \) and \( H\left( {p + \delta }\right) < H\left( p\right) + \epsilon /2 \) . For \( n \) large set \( m = n\left( {1 - H\left( p\right) - \epsilon }\right) \), guaranteeing the rate of transmission. Let \( f \) : \( \{ 0,1{\} }^{m} \rightarrow \{ 0,1{\} }^{n... | Yes |
Theorem 14.1.2 Let \( p \in \left( {0,{.5}}\right) \) be fixed. For \( \epsilon > 0 \) arbitrarily small there exists a Group Code with rate of transmission greater than \( 1 - H\left( p\right) - \epsilon \) and probability of incorrect transmission less than \( \epsilon \) . | Proof. For \( 1 \leq i \leq m \) let \( {u}_{i} \in \{ 0,1{\} }^{m} \) be that vector with a one in position \( i \) , all other entries zero. Let \( f\left( {u}_{1}\right) ,\ldots, f\left( {u}_{m}\right) \) be chosen randomly and independently and then extend \( f \) by setting\n\n\[ f\left( {{\epsilon }_{1}{u}_{1} + ... | Yes |
Corollary 14.2.2 If\n\n\[ n > \frac{{2}^{q}}{\mathop{\sum }\limits_{{i = 0}}^{k}\left( \begin{matrix} q \\ i \end{matrix}\right) } \]\n\nthen Carole wins the \( \left( {n, q, k}\right) \) -Liar Game. | Proof.[Theorem 14.2.1] Fix a strategy for Paul. Now Carole plays randomly! That is, at each round, after Paul has selected a set \( S \) of chips Carole flips a coin - if it comes up heads she moves every chip not in \( S \) one position to the left and if it comes up tails she moves every chip in \( S \) one position ... | Yes |
Theorem 14.3.1 If \( \mathop{\sum }\limits_{i}{x}_{i}{2}^{-i} < 1 \) then Carole wins the \( \left( {{x}_{1},\ldots ,{x}_{k}}\right) \) -Tenure Game. | Proof. Fix a strategy for Paul. Now Carole plays randomly! That is, at each round, after Paul has selected a set \( S \) of chips Carole flips a coin - if it comes up heads she moves every chip not in \( S \) one position to the left and if it comes up tails she moves every chip in \( S \) one position to the left. For... | Yes |
Lemma 14.3.2 If a set of chips has weight at least one it may be split into two parts each of weight at least one half. | Proof. There must be two chips at some position \( i \), otherwise the weight is less than one. If there are two chips at position 1 simply split them. If there are two chips at position \( i > 1 \) glue them together, and consider them as one superchip at position \( i - 1 \) . Then the proof follows by induction on t... | No |
Theorem 14.3.3 If \( \sum {x}_{i}{2}^{-i} \geq 1 \) then Paul wins the \( \left( {{x}_{1},\ldots ,{x}_{k}}\right) \) -Tenure Game. | Proof. The initial weight is at least one. Applying the Lemma Paul splits the chips into two parts each of weight at least one half and sets \( S \) equal one of the parts. Carole moves all chips in one part one position to the left, doubling their weight, leaving a new position of weight at least one. Thus the weight ... | Yes |
Theorem 14.4.1 If \( \Pr \left\lbrack {\left| {S}_{n}\right| > \alpha }\right\rbrack < {n}^{-1} \) then \( \operatorname{VAL}\left( n\right) \leq \alpha \) . | Proof. Consider the game a win for Pusher if the final \( {\left| P\right| }_{\infty } > \alpha \) . Suppose Chooser announces that she will flip a fair coin each round to determine whether to reset \( P \) as \( P + v \) or \( P - v \) . Let \( {x}_{i} \) be the \( i \) -th coordinate for the final value of the positi... | Yes |
Lemma 14.6.1 Let \( X, Y \) and \( Z \) be three random variables taking values in \( S, T \) and \( U \), respectively. Then\n\n1. \( H\left( X\right) \leq {\log }_{2}\left| S\right| \) .\n\n2. \( H\left( {X, Y}\right) \geq H\left( X\right) \) . | ## Proof.\n\n1. Since the function \( \log z \) is concave it follows, by Jensen’s Inequality, that\n\n\[ H\left( X\right) = \mathop{\sum }\limits_{{i \in S}}P\left( {X = i}\right) \log \left( \frac{1}{P\left( {X = i}\right) }\right) \]\n\n\[ \leq \log \left( {\mathop{\sum }\limits_{{i \in S}}P\left( {X = i}\right) \fr... | Yes |
Proposition 14.6.2 Let \( X = \left( {{X}_{1},\ldots ,{X}_{n}}\right) \) be a random variable taking values in the set \( S = {S}_{1} \times {S}_{2} \times \ldots \times {S}_{n} \), where each of the coordinates \( {X}_{i} \) of \( X \) is a random variable taking values in \( {S}_{i} \). Then\n\n\[ H\left( X\right) \l... | Proof. This follows by induction from Lemma 14.6.1, part 3. ∎ | No |
Corollary 14.6.3 Let \( \mathcal{F} \) be a family of subsets of \( \{ 1,2,\ldots, n\} \) and let \( {p}_{i} \) denote the fraction of sets in \( \mathcal{F} \) that contain \( i \) . Then\n\n\[ \left| \mathcal{F}\right| \leq {2}^{\mathop{\sum }\limits_{{i = 1}}^{n}H\left( {p}_{i}\right) } \]\n\nwhere \( H\left( y\righ... | Proof. Associate each set \( F \in \mathcal{F} \) with its characteristic vector \( v\left( F\right) \), which is a binary vector of length \( n \) . Let \( X = \left( {{X}_{1},\ldots ,{X}_{n}}\right) \) be the random variable taking values in \( \{ 0,1{\} }^{n} \), where \( P\left( {X = v\left( F\right) }\right) = 1/\... | Yes |
Proposition 14.6.4 Let \( X = \left( {{X}_{1},\ldots ,{X}_{n}}\right) \) and \( S \) be as above. If \( \mathcal{G} \) is a family of subsets of \( \{ 1,\ldots, n\} \) and each \( i \in \{ 1,\ldots, n\} \) belongs to at least \( k \) members of \( \mathcal{G} \) then\n\n\[ \n{kH}\left( X\right) \leq \mathop{\sum }\limi... | Proof. We apply induction on \( k \) . For \( k = 1 \), replace each set \( G \in \mathcal{G} \) by a subset of it to obtain a family \( {\mathcal{G}}^{\prime } \) whose members form a partition of \( \{ 1,\ldots, n\} \) . By Lemma 14.6.1, part 2, \( \mathop{\sum }\limits_{{G \in \mathcal{G}}}H\left( {X\left( G\right) ... | Yes |
Corollary 14.6.5 Let \( \mathcal{F} \) be a family of vectors in \( {S}_{1} \times {S}_{2}\ldots \times {S}_{n} \) . Let \( \mathcal{G} = \) \( \left\{ {{G}_{1},{G}_{2},\ldots {G}_{m}}\right\} \) be a collection of subsets of \( N = \{ 1,2,\ldots, n\} \), and suppose that each element \( i \in N \) belongs to at least ... | Proof. Let \( X = \left( {{X}_{1},\ldots ,{X}_{n}}\right) \) be the random variable taking values in \( \mathcal{F} \), where \( P\left( {X = F}\right) = \frac{1}{\left| \mathcal{F}\right| } \) for all \( F \in \mathcal{F} \) . By Proposition 14.6.4\n\n\[ \n{kH}\left( X\right) \leq \mathop{\sum }\limits_{{i = 1}}^{m}H\... | Yes |
Corollary 14.6.7 [Chung et al. (1986) ] Let \( N \) be a finite set, and let \( \mathcal{F} \) be a family of subsets of \( N \) . Let \( \mathcal{G} = \left\{ {{G}_{1},\ldots {G}_{m}}\right\} \) be a collection of subsets of \( N \), and suppose that each element of \( S \) belongs to at least \( k \) members of \( \m... | \[ {\left| \mathcal{F}\right| }^{k} \leq \mathop{\prod }\limits_{{i = 1}}^{m}\left| {\mathcal{F}}_{i}\right| \] | Yes |
Corollary 14.6.8 Let \( \\mathcal{F} \) be a family of graphs on the labeled set of vertices \( \\{ 1,2,\\ldots t\\} \), and suppose that for any two members of \( \\mathcal{F} \) there is a triangle contained in both of them. Then\n\n\[ \n\\left| \\mathcal{F}\\right| < \\frac{1}{4}{2}^{\\left( \\begin{array}{l} t \\\\... | Proof. Let \( N \) be the set of all \( \\left( \\begin{array}{l} t \\\\ 2 \\end{array}\\right) \) unordered pairs of vertices in \( T = \\{ 1,2\\ldots, t\\} \) , and consider \( \\mathcal{F} \) as a family of subsets of \( N \) . Let \( \\mathcal{G} \) be the family of all subsets of \( N \) consisting of the edge-set... | Yes |
Theorem 15.2.1 Suppose \( n = {2}^{k} - 1 \) and \( d = {2t} + 1 \) . Then there exists a symmetric probability space \( \Omega \) of size \( 2{\left( n + 1\right) }^{t} \) and \( d \) -wise independent random variables \( {y}_{1},\ldots ,{y}_{n} \) over \( \Omega \) each of which takes the values 0 and 1 with probabil... | The space and the variables are explicitly constructed, given a representation of the field \( F = {GF}\left( {2}^{k}\right) \) as a \( k \) -dimensional algebra over \( {GF}\left( 2\right) \) . Proof. Let \( {x}_{1},\ldots ,{x}_{n} \) be the \( n \) nonzero elements of \( F \), represented as column-vectors of length ... | Yes |
Lemma 15.2.2 Any set of \( d = {2t} + 1 \) columns of \( H \) is linearly independent over \( {GF}\left( 2\right) \) . | Proof. Let \( J \subset \{ 1,2,\ldots, n\} \) be a subset of cardinality \( \left| J\right| = {2t} + 1 \) of the set of indices of the columns of \( H \) . Suppose that \( \mathop{\sum }\limits_{{j \in J}}{z}_{j}{H}_{j} = 0 \), where \( {H}_{j} \) denotes the \( j \) -th column of \( H \) and \( {z}_{j} \in {GF}\left( ... | Yes |
Proposition 15.2.3 If the random variables \( {y}_{1},\ldots ,{y}_{n} \) over the sample space \( \Omega \) are \( d \) -wise independent and none of them is almost constant then \( \left| \Omega \right| \geq m\left( {n, d}\right) \) . | Proof. Clearly we may assume that the expected value of each \( {y}_{j} \) is 0 (since otherwise we can replace \( {y}_{j} \) by \( {y}_{j} - E\left( {y}_{j}\right) \) ). For each subset \( S \) of \( \{ 1,\ldots, n\} \), define \( {\alpha }_{S} = \mathop{\prod }\limits_{{j \in S}}{y}_{j} \) . Observe that since no \( ... | Yes |
Theorem 1 The crossing number of any simple graph \( G = \left( {V, E}\right) \) with \( \left| E\right| \geq 4\left| V\right| \) is at least \( \frac{{\left| E\right| }^{3}}{{64}{\left| V\right| }^{2}} \) . | Proof. By Euler’s formula any simple planar graph with \( n \) vertices has at most \( {3n} - 6 \) edges, implying that the crossing number of any simple graph with \( n \) vertices and \( m \) edges is at least \( m - \left( {{3n} - 6}\right) > m - {3n} \) . Let \( G = \left( {V, E}\right) \) be a graph with \( \left|... | Yes |
Theorem 2 Let \( P \) be a set of \( n \) distinct points in the plane, and let \( L \) be a set of \( m \) distinct lines. Then, the number of incidences between the members of \( P \) and those of \( L \) (that is, the number of pairs \( \left( {p, l}\right) \) with \( p \in P, l \in L \) and \( p \in l \) ) is at mo... | Proof. Denote the number of incidences by \( I \) . Let \( G = \left( {V, E}\right) \) be the graph whose vertices are all members of \( P \), where two are adjacent if and only if they are consecutive points of \( P \) on some line in \( L \) . Clearly, \( \left| V\right| = n \) and \( \left| E\right| = I - m \) . Not... | Yes |
Theorem 3 For any three sets \( A, B \) and \( C \) of \( s \) real numbers each,\n\n\[ \left| {A \cdot B + C}\right| = \left| {\{ {ab} + c : a \in A, b \in B, c \in C\} }\right| \geq \Omega \left( {s}^{3/2}\right) . \]\n | Proof. Put \( R = A \cdot B + C,\left| R\right| = r \) and define\n\n\[ P = \{ \left( {a, t}\right) : a \in A, t \in R\} ,\;L = \{ y = {bx} + c : b \in B, c \in C\} . \]\n\nThus \( P \) is a set of \( n = {sr} \) points in the plane, \( L \) is a set of \( m = {s}^{2} \) lines in the plane, and each line \( y = {bx} + ... | Yes |
Proposition 1 Let \( G = \left( {V, E}\right) \) be a triangle-free graph on \( n \) vertices with maximum degree at most \( d \geq 1 \) . Then\n\n\[ \alpha \left( G\right) \geq \frac{n\log d}{8d} \]\n\nwhere the logarithm here and in what follows is in base 2. | Proof. If, say, \( d < {16} \) the result follows from the trivial bound \( \alpha \left( G\right) \geq n/\left( {d + 1}\right) \) and hence we may and will assume that \( d \geq {16} \) . Let \( W \) be a random independent set of vertices in \( G \), chosen uniformly among all independent sets in \( G \) . For each v... | Yes |
Theorem 2 [Ajtai et al. (1980) ] There exists an absolute constant \( b \) such that \( r\left( {3, k}\right) \leq b{k}^{2}/\log k \) for every \( k > 1 \) . | Proof. Let \( G = \left( {V, E}\right) \) be a triangle-free graph on \( 8{k}^{2}/\log k \) vertices. If \( G \) has a vertex of degree at least \( k \) then its neighborhood contains an independent set of size \( k \) . Otherwise, by proposition 1 above, \( G \) contains an independent set of size at least \( \frac{8{... | Yes |
Proposition 1.1 Let \( {\left( {X}_{n}\right) }_{n \geq 1} \) be a sequence of real random variables such that, for every \( n \geq 1,{X}_{n} \) follows the \( \mathcal{N}\left( {{m}_{n},{\sigma }_{n}^{2}}\right) \) -distribution. Suppose that \( {X}_{n} \) converges in \( {L}^{2} \) to X. Then:\n\n(i) The random varia... | (i) The convergence in \( {L}^{2} \) implies that \( {m}_{n} = E\left\lbrack {X}_{n}\right\rbrack \) converges to \( E\left\lbrack X\right\rbrack \) and \( {\sigma}_{n}^{2} = \) \( \operatorname{var}\left( {X}_{n}\right) \) converges to \( \operatorname{var}\left( X\right) \) as \( n \rightarrow \infty \) . Then, setti... | Yes |
Proposition 1.2 Under the preceding assumptions, the random variables \( {X}_{1},\ldots ,{X}_{d} \) are independent if and only if the covariance matrix \( {\left( \operatorname{cov}\left( {X}_{j},{X}_{k}\right) \right) }_{1 \leq j, k \leq d} \) is diagonal or equivalently if and only if \( {q}_{X} \) is of diagonal fo... | Proof If the random variables \( {X}_{1},\ldots ,{X}_{d} \) are independent, the covariance matrix \( {\left( \operatorname{cov}\left( {X}_{j},{X}_{k}\right) \right) }_{j, k = 1,\ldots d} \) is diagonal. Conversely, if this matrix is diagonal, we have for every \( u = \mathop{\sum }\limits_{{j = 1}}^{d}{u}_{j}{e}_{j} \... | Yes |
Proposition 1.7 If \( {\left( {X}_{t}\right) }_{t \in T} \) is a Gaussian process, the closed linear subspace of \( {L}^{2} \) spanned by the variables \( {X}_{t}, t \in T \), is a Gaussian space, which is called the Gaussian space generated by the process \( X \) . | Proof It suffices to observe that an \( {L}^{2} \) -limit of centered Gaussian variables is still centered Gaussian, by Proposition 1.1. | No |
Theorem 1.9 Let \( H \) be a centered Gaussian space and let \( {\left( {H}_{i}\right) }_{i \in I} \) be a collection of linear subspaces of \( H \) . Then the subspaces \( {H}_{i}, i \in I \), are (pairwise) orthogonal in \( {L}^{2} \) if and only the \( \sigma \) -fields \( \sigma \left( {H}_{i}\right), i \in I \), a... | Proof Suppose that the \( \sigma \) -fields \( \sigma \left( {H}_{i}\right) \) are independent. Then, if \( i \neq j \), if \( X \in {H}_{i} \) and \( Y \in {H}_{j} \) ,\n\n\[ E\left\lbrack {XY}\right\rbrack = E\left\lbrack X\right\rbrack E\left\lbrack Y\right\rbrack = 0, \]\n\nso that the linear spaces \( {H}_{i} \) a... | Yes |
Corollary 1.10 Let \( H \) be a (centered) Gaussian space and let \( K \) be a closed linear subspace of \( H \) . Let \( {p}_{K} \) denote the orthogonal projection onto \( K \) in the Hilbert space \( {L}^{2} \), and let \( X \in H \) . (i) We have \[ E\left\lbrack {X \mid \sigma \left( K\right) }\right\rbrack = {p}_... | ## Proof (i) Let \( Y = X - {p}_{K}\left( X\right) \) . Then \( Y \) is orthogonal to \( K \) and, by Theorem \( {1.9}, Y \) is independent of \( \sigma \left( K\right) \) . Then, \[ E\left\lbrack {X \mid \sigma \left( K\right) }\right\rbrack = E\left\lbrack {{p}_{K}\left( X\right) \mid \sigma \left( K\right) }\right\r... | Yes |
Theorem 1.11 Let \( \Gamma \) be a symmetric function of positive type on \( T \times T \) . There exists, on an appropriate probability space \( \left( {\Omega ,\mathcal{F}, P}\right) \), a centered Gaussian process whose covariance function is \( \Gamma \) . | Example Consider the case \( T = \mathbb{R} \) and let \( \mu \) be a finite measure on \( \mathbb{R} \), which is also symmetric (i.e. \( \mu \left( {-A}\right) = \mu \left( A\right) \) ). Then set, for every \( s, t \in \mathbb{R} \) ,\n\n\[ \Gamma \left( {s, t}\right) = \int {\mathrm{e}}^{\mathrm{i}\xi \left( {t - s... | Yes |
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