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Proposition 1.10.3. A nonempty space \( X \) has dimension 0 if and only if every point \( p \in X \) and every closed set \( C \subseteq X \) such that \( p \notin C \) can be separated by open sets. | Proof: First suppose that \( X \) has dimension 0 according to our original definition. Certainly \( X \smallsetminus C \) is a neighborhood of \( p \) . So there is a set \( V \) with \( p \in V \subseteq X \smallsetminus C \) and \( V \) is both open and closed. But \( V \cap C = \varnothing \), so \( p \) and \( C \... | No |
Lemma 1.10.4. A connected, 0-dimensional space consists of just one point. | Proof: Suppose instead that the space contains two points. Then Proposition 1.10.3 tells us that these points are separated. So the space is in fact disconnected, and that is a contradiction. | Yes |
Proposition 1.10.5. A 0-dimensional space is totally disconnected. | Proof: This follows from 1.10.2 and 1.10.3. | No |
Proposition 1.11.1. The Cantor set \( C \) has zero length, in the sense that the complementary set \( \left\lbrack {0,1}\right\rbrack \smallsetminus C \) has length 1 . | Proof: In the construction of \( {S}_{1} \), we removed from the unit interval one interval of length \( {3}^{-1} \) . In constructing \( {S}_{2} \), we further removed two intervals of length \( {3}^{-2} \) . In constructing \( {S}_{j} \), we removed \( {2}^{j - 1} \) intervals of length \( {3}^{-j} \) . Thus the tota... | Yes |
Theorem 1.11.3. Let \( C \) be the Cantor set and define\n\n\[ K = \{ x + y : x \in C, y \in C\} . \]\n\nThen \( K = \left\lbrack {0,2}\right\rbrack \) . | Proof: We sketch the proof.\n\nSince \( C \subseteq \left\lbrack {0,1}\right\rbrack \) it is clear that \( K \subseteq \left\lbrack {0,2}\right\rbrack \) . For the reverse inclusion, fix an element \( t \in \left\lbrack {0,2}\right\rbrack \) . Our job is to find two elements \( c \) and \( d \) in \( C \) such that \( ... | No |
Theorem 1.12.2. Let \( \left( {X, d}\right) \) be a metric space. A set \( K \subseteq X \) is compact if and only if every sequence \( \left\{ {a}_{j}\right\} \subseteq K \) has a convergent subsequence (we call this last condition sequential compactness). | Proof: Suppose that \( K \subseteq X \) is compact according to our usual definition and let \( \left\{ {a}_{j}\right\} \) be a sequence in \( K \) . Seeking a contradiction, we assume that \( \left\{ {a}_{j}\right\} \) does not have a convergent subsequence. Then each element \( k \in K \) has a neighborhood \( {U}_{k... | No |
Proposition 1.12.3. Let \( \left( {X, d}\right) \) be a metric space and \( f : X \rightarrow \mathbb{R} \) a function. Then \( f \) is continuous if and only if, for each \( x \in X \) and each \( \varepsilon > 0 \), there is a \( \delta > 0 \) such that if \( d\left( {x, t}\right) < \delta \) then \( \left| {f\left( ... | Proof: This proof is the same as that of Proposition 1.3.3. | No |
Proposition 1.12.4. In a metric space \( \left( {X, d}\right) \), the metric function \( d \) is continuous. | Proof: In fact\n\n\[ d\left( {x, y}\right) \leq d\left( {x, z}\right) + d\left( {z, y}\right) \]\n\nhence\n\n\[ d\left( {x, y}\right) - d\left( {x, z}\right) \leq d\left( {z, y}\right) = d\left( {y, z}\right) . \]\n\nBy symmetry,\n\n\[ d\left( {x, z}\right) - d\left( {x, y}\right) \leq d\left( {z, y}\right) . \]\n\nIt ... | Yes |
Theorem 1.13.1. Let \( \\left( {X,\\mathcal{U}}\\right) \) be a \( {T}_{1} \) topological space. Then the following are equivalent:\n\n(a) \( X \) is regular and second countable. \( {}^{4} \)\n\n(b) \( X \) is separable \( {}^{5} \) and metrizable.\n\n(c) \( X \) can be embedded as a subspace of the Hilbert cube \( {I... | Proof: We divide the proof into three natural parts.\n\n(a) \( \\Rightarrow \) (c): Let \( \\mathcal{B} \) be a countable basis for \( X \) and define \( \\mathcal{C} = \\{ \\left( {U, V}\\right) \) : \( U, V \\in \\mathcal{B},\\bar{U} \\subseteq V\\} \) . Of course \( \\mathcal{C} \) is a countable set. We know from\n... | No |
Proposition 1.14.3. There is a continuous, real-valued function on the interval \( I = \left\lbrack {0,1}\right\rbrack \) that is nowhere differentiable on \( I \) . | Proof of Proposition 1.14.3: Let \( C\left( I\right) \) be the space of all real, continuous functions on the interval \( I \) . Equip this space with the uniform metric:\n\n\[ d\left( {f, g}\right) = \mathop{\max }\limits_{{x \in I}}\left| {f\left( x\right) - g\left( x\right) }\right| .\n\]\n\nDefine \( \mathcal{F} \)... | Yes |
Lemma 1.15.1. Let \( X \) be a compact metric space and \( {U}_{{\alpha }_{1}},{U}_{{\alpha }_{2}},\ldots ,{U}_{{\alpha }_{k}} \) a finite open cover. There is a number \( \varepsilon > 0 \) -called the Lebesgue number of the covering-so that any ball \( B\left( {x,\varepsilon }\right) \) must lie entirely inside some ... | Proof: There are many different ways to prove this result. We provide a proof by contradiction.\n\nSuppose that the assertion is not true. Then there are points \( {x}_{\ell } \in X \) and numbers \( {\varepsilon }_{\ell } \searrow 0 \) so that, for each \( \ell \), the ball \( B\left( {{x}_{\ell },{\varepsilon }_{\ell... | Yes |
Proposition 2.4.1. Any second countable space is separable. | Proof: Let \( \left( {X,\mathcal{U}}\right) \) be the topological space. Let \( \left\{ {U}_{j}\right\} \) be a countable basis for the topology on \( X \) . Select a point \( {p}_{j} \in {U}_{j} \) for each \( j \) . We claim that the countable set \( \left\{ {p}_{j}\right\} \) is dense.\n\nFor let \( x \in X \) be ar... | Yes |
Proposition 2.4.2. Let \( X \) be a separable metric space. Then \( X \) is second countable. | Proof: Let \( \left\{ {p}_{j}\right\} \) be a countable dense set in \( X \) . The countable base for the topology will be all metric balls with center \( {p}_{j} \) for some \( j \) and rational radius. To see this, let \( U \) be any open set and let \( x \in U \) . We need to produce one of the indicated balls that ... | Yes |
Proposition 2.5.1. The compactified space \( {X}^{ * } \) in Definition 2.5.3 is compact. | Proof: Let \( \mathcal{U} = {\left\{ {U}_{\alpha }\right\} }_{\alpha \in A} \) be an open covering of \( {X}^{ * } \), with the open sets taken from the topology \( {\mathcal{U}}^{ * } \) . Then there is at least one set that covers \( p \) . Select one of them and call it \( {U}_{{\alpha }_{0}} \) . All the sets \( {U... | Yes |
Theorem 2.7.1. Let \( X \) be a set and \( \mathcal{D} \) a uniformity on \( X \) . For each \( x \in X \) , the collection \( {\mathcal{U}}_{x} \equiv \{ D\left\lbrack x\right\rbrack : D \in \mathcal{D}\} \) forms a neighborhood basis at \( x \) . Thus the uniformity induces a topology on \( X \) . This topology is Ha... | Proof: Certainly \( x \in D\left\lbrack x\right\rbrack \) for each \( x \) . Also \( {D}_{1}\left\lbrack x\right\rbrack \cap {D}_{2}\left\lbrack x\right\rbrack = \left( {{D}_{1} \cap }\right. \) \( \left. {D}_{2}\right) \left\lbrack x\right\rbrack \), hence the intersection of neighborhoods is a neighborhood. If \( D\l... | Yes |
Proposition 2.9.1. Let \( X, Y \) be Euclidean spaces. Let \( E \subseteq X \) and \( F \subseteq Y \) be bounded, open sets. Suppose that \( f : E \rightarrow F \) is proper: If \( \left\{ {e}_{j}\right\} \subseteq E \) satisfies \( {e}_{j} \rightarrow \partial E \) then \( f\left( {e}_{j}\right) \rightarrow \partial ... | Proof: If not then there are \( {e}_{j} \in E \) with \( {e}_{j} \rightarrow \partial E \) but \( f\left( {e}_{j}\right) \nrightarrow \partial F \) . It follows that there is a compact set \( K \subseteq F \) so that \( f\left( {e}_{j}\right) \in K \) for all \( j \) . But then \( {f}^{-1}\left( K\right) \) is compact,... | Yes |
Proposition 2.9.2. Let \( X, Y \) be topological spaces and let \( f : X \rightarrow Y \) be a homeomorphism. Then \( f \) is proper. | Proof: Now \( {f}^{-1} : Y \rightarrow X \) is a continuous mapping. If \( K \subseteq Y \) is compact then of \( {f}^{-1}\left( K\right) \) must also be compact. So \( f \) is proper. | Yes |
Proposition 2.9.3. Every continuous mapping \( f \) from a compact space \( X \) to a Hausdorff space \( Y \) is both proper and closed. | Proof: The closedness is clear, for if \( E \subseteq X \) is closed then it is compact. Hence \( f\left( E\right) \) is compact and, since \( Y \) is Hausdorff, \( f\left( E\right) \) is closed.\n\nFor the properness, let \( F \subseteq Y \) be compact. Then \( F \) is certainly closed. Since \( f \) is continuous, we... | Yes |
Proposition 2.9.4. Let \( X \) be a topological space. Then \( X \) is compact if and only if the map of \( X \) to a single-point space \( Z \) is proper. | Proof: Let \( Z = \{ z\} \) . Suppose that \( f : X \rightarrow Z \) is proper. Certainly \( Z \) is compact, so \( {f}^{-1}\left( Z\right) \) is compact. But \( {f}^{-1}\left( Z\right) = X \) . That proves one direction.\n\nNow suppose that \( X \) is compact. Let \( f : X \rightarrow Z \) . Let \( F \) be a compact s... | Yes |
Theorem 2.10.1. Let \( X \) be a paracompact topological space. Let \( \mathcal{U} = \) \( {\left\{ {U}_{\alpha }\right\} }_{\alpha \in A} \) be an open cover of \( X \) . Then there is a partition of unity \( \left\{ {\varphi }_{\alpha }\right\} \) subordinate to \( \mathcal{U} \) . | Proof: For simplicity we shall treat only the case when \( X \) is a metric space with metric \( d \) . Let \( \mathcal{V} = {\left\{ {V}_{\beta }\right\} }_{\beta \in B} \) be a refinement of \( \mathcal{U} \) that is locally finite and which still covers \( X \) . For each \( \beta \in B \), define\n\n\[ \n{\psi }_{\... | Yes |
Theorem 3.2.2. Let \( \left( {X,\mathcal{U}}\right) \) be a topological space. Then\n\n(a) A point \( s \) is an accumulation point of a subset \( A \) of \( X \) if and only if there is a net in \( A \smallsetminus \{ s\} \) which converges to \( s \) . | Proof: If \( s \) is an accumulation point of \( A \) then, for each neighborhood \( U \) of \( s \), there is a point \( {t}_{U} \) of \( A \) that belongs to \( U \smallsetminus \{ s\} \) . The family \( \mathcal{U} \) of all neighborhoods \( U \) of \( s \) is directed by \( \subseteq \) . If \( U \) and \( V \) are... | Yes |
Theorem 3.2.3. A topological space \( \left( {X,\mathcal{U}}\right) \) is Hausdorff if and only if each net in the space converges to at most one point. | Proof: Let \( X \) be Hausdorff and let \( s, t \in X \) be distinct points. Then there are disjoint neighborhoods \( U, V \) of \( s, t \) respectively. Since a net cannot eventually be in each of two disjoint neighborhoods, it is clear that no net in \( X \) converges to both points \( s \) and \( t \) .\n\nFor the c... | Yes |
Proposition 4.2.1. Let \( \mathcal{E} \) be a family of functions from a set \( S \) to \( \mathbb{R} \). Then \( \mathcal{E} \) is compact with respect to the topology of pointwise convergence provided that (a) The set \( \mathcal{E} \) is pointwise closed in \( {\mathbb{R}}^{S} \), (b) for each point \( s \in S \), t... | Proof: The family \( \mathcal{E} \) is a subfamily of \( {\mathbb{R}}^{S} \), which is also contained in \( { \times }_{s \in S}\overline{\mathcal{E}}\left( s\right) \). If condition (b) holds, then the product is a compact subset of \( {\mathbb{R}}^{S} \) by Tychanoff’s theorem. If \( \mathcal{E} \) is pointwise close... | Yes |
Proposition 4.2.2. Let \( \mathcal{E} \) be a family of functions on the set \( S \), with values in \( \mathbb{R} \) . Let \( A \subseteq S \) . The family \( \mathcal{E} \) with the topology of pointwise convergence on \( A \) is a Hausdorff space if and only if \( A \) distinguishes members of \( \mathcal{E} \) . | Proof: The product space \( {\mathbb{R}}^{A} \) is Hausdorff. Of course \( \mathcal{E} \) with the topology of pointwise convergence on \( A \) is Hausdorff if and only if the map that takes \( f \in \mathcal{E} \) to its restriction to the domain \( A \) is one-to-one. This will hold if and only if \( A \) distinguish... | Yes |
Proposition 4.3.1. The compact-open topology \( \mathcal{C} \) contains the topology \( \mathcal{P} \) of pointwise convergence. The space \( \left( {\mathcal{E},\mathcal{C}}\right) \) is a Hausdorff space provided that the range space \( Y \) is Hausdorff. | Proof: For each \( x \in X \) and each open subset \( U \subseteq Y \), the set\n\n\[ \mathcal{W}\left( {\{ x\}, U}\right) = \{ f : f\left( x\right) \in U\} \]\n\n(4.3.1.1)\n\nbelongs to \( \mathcal{C} \) because \( \{ x\} \) is compact. Therefore \( \mathcal{P} \subseteq \mathcal{C} \) because the family of all sets o... | Yes |
Theorem 4.4.1. Let \( \mathcal{E} \) be the family of all functions from a set \( X \) to a uniform space \( \left( {Y,\mathcal{V}}\right) \) . Let \( \mathcal{U} \) be the uniformity of uniform convergence on Y. Then\n\n(a) The uniformity \( \mathcal{U} \) is generated by the family of all pseudometrics of the form \(... | Proof: For part (a), if \( d \) is a bounded member of the gage of \( \mathcal{V} \), then the family of all sets of the form \( \{ \left( {y, z}\right) : d\left( {y, z}\right) \leq r\}, r > 0 \), is a basis for \( \mathcal{V} \) . This is so because if \( e \) is a pseudometric then the pseudometric \( {d}^{ * } = \mi... | Yes |
Theorem 4.4.2. Let \( \mathcal{E} \) be the family of all continuous functions from a topological space \( X \) to a uniform space \( \left( {Y,\mathcal{V}}\right) \) . Let \( \mathcal{U} \) be the uniformity of uniform convergence. Then\n\n(a) The family \( \mathcal{E} \) is closed in the space of all functions from \... | Proof: We prove part (a) indirectly by showing that the set of all noncontinuous functions is an open subset of the space \( \mathcal{G} \) of all functions from \( X \) to \( Y \) . If \( f \) is not continuous at a point \( x \in X \) then there is a member \( V \in \mathcal{V} \) such that \( {f}^{-1}\left\lbrack {V... | Yes |
Proposition 4.5.2. If the family \( \mathcal{E} \) of functions from \( X \) to \( Y \) is equicontinuous at \( x \), then the closure of \( \mathcal{E} \) relative to the topology \( \mathcal{P} \) of pointwise convergence is also equicontinuous at \( x \) . | Proof: Let \( x \in X \) and \( U \) a neighborhood of \( x \) . Let \( V \) be a member of the uniformity of \( Y \) that is a closed set. Then the class of all functions \( f \) that satisfy \( f\left\lbrack U\right\rbrack \subseteq V\left\lbrack {f\left( x\right) }\right\rbrack \) is closed relative to the topology ... | No |
Proposition 4.5.3. Let \( \mathcal{E} \) be an equicontinuous family of functions. Then the topology of pointwise convergence on \( \mathcal{E} \) is jointly continuous and therefore coincides with the topology of uniform convergence on compact sets. | Proof: We want to show that the map of \( \mathcal{E} \times X \rightarrow Y \) given by \( \left( {f, x}\right) \mapsto \) \( f\left( x\right) \) is continuous at a point \( \left( {f, x}\right) \) . Let \( V \) be a member of the uniformity of \( Y \) and let \( U \) be a neighborhood of \( x \) such that \( g\left\l... | Yes |
Theorem 4.5.4. Let \( \mathcal{E} \) be a family of functions from a topological space \( X \) to a uniform space \( \left( {Y,\mathcal{V}}\right) \) . If \( \mathcal{E} \) is compact relative to a jointly continuous topology, then it is equicontinuous. | Proof: Let \( x \in X \) be a fixed piont and \( V \) a symmetric member of \( \mathcal{V} \) . If we can show that there is a neighborhood \( U \) of \( x \) such that \( g\left\lbrack U\right\rbrack \subseteq V \circ V\left\lbrack {g\left( x\right) }\right\rbrack \) for each \( g \) in \( \mathcal{E} \), then the res... | Yes |
Theorem 4.5.5. Let \( \mathcal{C} \) be the family of all continuous functions from a regular, locally compact topological space \( X \) to a Hausdorff uniform space \( \left( {Y,\mathcal{V}}\right) \) . Assume that \( \mathcal{C} \) has the topology of uniform convergence on compact sets. Then a subfamily \( \mathcal{... | Proof: This is immediate from what went before-especially 4.5.3, 4.5.4. | No |
Lemma 4.6.2. Let \( {\psi }_{j} \) be a sequence of continuous functions on the interval \( \left\lbrack {-1,1}\right\rbrack \) with the following properties:\n\n(i) \( {\psi }_{j}\left( x\right) \geq 0 \) for all \( x \) ,\n\n(ii) \( {\int }_{-1}^{1}{\psi }_{j}\left( x\right) {dx} = 1 \) for each \( j \) ,\n\n(iii) Fo... | Proof: By multiplying \( f \) by a constant we may assume that \( \sup \left| f\right| = \) 1. Let \( \varepsilon > 0 \) . Since \( f \) is uniformly continuous on the interval \( \left\lbrack {0,1}\right\rbrack \) we may choose a \( \delta > 0 \) such that if \( \left| {x - t}\right| < \delta \) then \( \left| {f\left... | Yes |
Lemma 4.6.3. Define \( {\psi }_{j}\left( t\right) = {k}_{j} \cdot {\left( 1 - {t}^{2}\right) }^{j} \), where the positive constants \( {k}_{j} \) are chosen so that \( {\int }_{-1}^{1}{\psi }_{j}\left( t\right) {dt} = 1 \) . Then the functions \( {\psi }_{j} \) satisfy the properties (i)-(iii) of the last lemma. | Proof: Property (ii) is true by design. Property (i) is obvious. In order to verify property (iii), we need to estimate the size of \( {k}_{j} \). We have \[ {\int }_{-1}^{1}{\left( 1 - {t}^{2}\right) }^{j}{dt} = 2 \cdot {\int }_{0}^{1}{\left( 1 - {t}^{2}\right) }^{j}{dt} \] \[ \geq 2 \cdot {\int }_{0}^{1/\sqrt{j}}{\le... | Yes |
Let \( K \) be the complex shown in Fig. 8.5. The vertices \( {v}_{1},{v}_{2},{v}_{3},{v}_{4} \) generate \( {Z}_{0}\left( K\right) = {C}_{0}\left( K\right) \), and \( {C}_{1}\left( K\right) \) can be thought of as the free abelian group generated by the oriented 1-simplexes \( \left( {{v}_{1},{v}_{2}}\right) \) , \( \... | The group \( {Z}_{1}\left( K\right) \) is generated by the elementary 1-cycles, and by inspection there are six such, namely\n\n\[ \n{z}_{1} = \left( {{v}_{1},{v}_{2}}\right) + \left( {{v}_{2},{v}_{4}}\right) + \left( {{v}_{4},{v}_{1}}\right) \]\n\n\[ \n{z}_{2} = \left( {{v}_{2},{v}_{3}}\right) + \left( {{v}_{3},{v}_{4... | Yes |
If two vertices \( v, w \) of a complex \( K \) lie in the same component of \( \left| K\right| \), then they are homologous. | For we can join \( v \) to \( w \) by an edge path \( v{v}_{1}{v}_{2}\ldots {v}_{k}w \), in which no two consecutive vertices are equal, and then check that \( w - v \) is the boundary of the 1-chain \( \left( {v,{v}_{1}}\right) + \left( {{v}_{1},{v}_{2}}\right) + \ldots + \left( {{v}_{k}, w}\right) \). | Yes |
Let \( K \) be a triangulation of the torus. If we orient all the 2-simplexes of \( K \) compatibly, take their sum, and compute the boundary of this sum, then each edge of the triangulation occurs exactly twice in the result, once with each of its two possible orientations. So we have a two-dimensional cycle. It is el... | (For suppose the oriented triangle \( \left( {a, b, c}\right) \) occurs in a 2-cycle with coefficient \( \lambda \) , then \( \lambda \left( {b, c}\right) \) automatically appears in its boundary. Now the edge spanned by \( b \) and \( c \) lies in precisely one other triangle of \( K \) whose third vertex we denote by... | Yes |
Suppose we have a complex \( K \) which is a cone, in other words \( K \) is isomorphic to a complex of the form \( {CL} \) where the dimension of \( L \) is one less than that of \( K \) . Let \( v \) be the unique vertex of \( K \) which does not lie in \( L \), usually called the apex of \( K \). | A cone is always connected, so \( {H}_{0}\left( K\right) \cong \mathbb{Z} \), by theorem (8.2). Now assume \( q > 0 \) and define a homomorphism \( d : {C}_{q}\left( K\right) \rightarrow {C}_{q + 1}\left( K\right) \) as follows. If \( \sigma = \left( {{v}_{0},\ldots ,{v}_{q}}\right) \) is an oriented \( q \) -simplex o... | No |
Let \( {\Delta }^{n + 1} \) denote an \( \left( {n + 1}\right) \) -simplex, \( n > 0 \), together with all its faces, thought of as a simplicial complex, and let \( {\sum }^{n} \) denote those simplexes which lie in the boundary of \( {\Delta }^{n + 1} \) . So \( \left| {\sum }^{n}\right| \) is homeomorphic to \( {S}^{... | But \( {\Delta }^{n + 1} \) is a cone, so by Example 4 we have \( {H}_{0}\left( {\sum }^{n}\right) \cong \mathbb{Z} \) and \( {H}_{q}\left( {\sum }^{n}\right) = 0 \) for \( 1 \leq q \leq n - 1 \) . (Remember we have assumed \( n > 0.{\sum }^{0} \) consists of two points and so \( {H}_{0}\left( {\sum }^{0}\right) \cong ... | Yes |
If \( \left| \mathrm{K}\right| \) is connected, abelianizing its fundamental group gives the first homology group of \( \mathbf{K} \) . | To show \( \phi \) is onto, we need only prove that the homology class of each elementary 1-cycle lies in the image of \( \phi \) . Now an elementary 1-cycle is just an oriented simple edge loop thought of as the sum of its oriented edges, say \( {z}_{1} = \left( {{w}_{1},{w}_{2}}\right) + \left( {{w}_{2},{w}_{3}}\righ... | Yes |
Let \( u \) be a continuous complex-valued function on \( \left\lbrack {0,1}\right\rbrack \). We can associate with it a multiplication operator \( {M}_{u} \) on \( C\left\lbrack {0,1}\right\rbrack \) as follows. Given a function \( f \), let \( {M}_{u}\left( f\right) \) be the function that takes \( t \) to \( u\left(... | (i) Let \( u \) be the constant function \( u\left( t\right) \equiv k \). Then evidently \( {M}_{u} \) has the single eigenvalue \( k \) and every (nonzero) function \( f \) in \( X \) is an eigenvector.\n\n(ii) Let \( u\left( t\right) = t \) for all \( t \). Suppose that the complex number \( \lambda \) is an eigenval... | Yes |
Theorem 1. If a \( 0 - 1 \) matrix \( A \) has only real positive eigenvalues, then those eigenvalues are all equal to 1 . | To prove this, let \( {\left\{ {\lambda }_{i}\right\} }_{i = 1}^{n} \) be the eigenvalues of \( A \) . Then\n\n\[ 1 \geq \frac{1}{n}\operatorname{trace}\left( A\right) \;\left( {\text{ since all }{A}_{i, i} \leq 1}\right) \]\n\n\[ = \frac{1}{n}\left( {{\lambda }_{1} + {\lambda }_{2} + \cdots + {\lambda }_{n}}\right) \]... | Yes |
Proposition 1.1.1 If \( \left( \begin{array}{l} n \\ k \end{array}\right) \cdot {2}^{1 - \left( \begin{array}{l} k \\ 2 \end{array}\right) } < 1 \) then \( R\left( {k, k}\right) > n \) . Thus \( R\left( {k, k}\right) > \left\lfloor {2}^{k/2}\right\rfloor \) for all \( k \geq 3 \) . | Proof. Consider a random two-coloring of the edges of \( {K}_{n} \) obtained by coloring each edge independently either red or blue, where each color is equally likely. For any fixed set \( R \) of \( k \) vertices, let \( {A}_{R} \) be the event that the induced subgraph of \( {K}_{n} \) on \( R \) is monochromatic (i... | Yes |
Theorem 1.2.1 If \( \left( \begin{array}{l} n \\ k \end{array}\right) {\left( 1 - {2}^{-k}\right) }^{n - k} < 1 \) then there is a tournament on \( n \) vertices that has the property \( {S}_{k} \) . | Proof. Consider a random tournament on the set \( V = \{ 1,\ldots, n\} \) . For every fixed subset \( K \) of size \( k \) of \( V \), let \( {A}_{K} \) be the event that there is no vertex which beats all the members of \( K \) . Clearly \( \Pr \left( {A}_{K}\right) = {\left( 1 - {2}^{-k}\right) }^{n - k} \) . This is... | Yes |
Theorem 1.2.2 Let \( G = \left( {V, E}\right) \) be a graph on \( n \) vertices, with minimum degree \( \delta > 1 \) . Then \( G \) has a dominating set of at most \( n\frac{1 + \ln \left( {\delta + 1}\right) }{\delta + 1} \) vertices. | Proof. Let \( p \in \left\lbrack {0,1}\right\rbrack \) be, for the moment, arbitrary. Let us pick, randomly and independently, each vertex of \( V \) with probability \( p \) . Let \( X \) be the (random) set of all vertices picked and let \( Y = {Y}_{X} \) be the random set of all vertices in \( V - X \) that do not h... | Yes |
Lemma 1.2.3 Let \( G = \left( {V, E}\right) \) be a graph with minimum degree \( \delta \) and let \( V = {V}_{1} \cup {V}_{2} \) be a cut of size smaller than \( \delta \) in \( G \) . Then every dominating set \( U \) of \( G \) has vertices in \( {V}_{1} \) and in \( {V}_{2} \) . | Proof. Suppose this is false and \( U \subseteq {V}_{1} \) . Choose, arbitrarily, a vertex \( v \in {V}_{2} \) and let \( {v}_{1},{v}_{2},\ldots ,{v}_{\delta } \) be \( \delta \) of its neighbors. For each \( i,1 \leq i \leq \delta \), define an edge \( {e}_{i} \) of the given cut as follows; if \( {v}_{i} \in {V}_{1} ... | Yes |
Proposition 1.3.1 [Erdős (1963a)] Every n-uniform hypergraph with less than \( {2}^{n - 1} \) edges has property \( B \) . Therefore \( m\left( n\right) \geq {2}^{n - 1} \) . | Proof. Let \( H = \left( {V, E}\right) \) be an \( n \) -uniform hypergraph with less than \( {2}^{n - 1} \) edges. Color \( V \) randomly by 2 colors. For each edge \( e \in E \), let \( {A}_{e} \) be the event that \( e \) is monochromatic. Clearly \( \Pr \left( {A}_{e}\right) = {2}^{1 - n} \) . Therefore\n\n\[ \Pr \... | Yes |
Theorem 1.3.3 If \( \mathcal{F} = {\left\{ \left( {A}_{i},{B}_{i}\right) \right\} }_{i = 1}^{h} \) is a \( \left( {k,\ell }\right) \) -system then \( h \leq \left( \begin{matrix} k + \ell \\ k \end{matrix}\right) \) . | Proof. Put \( X = \mathop{\bigcup }\limits_{{i = 1}}^{h}\left( {{A}_{i} \cup {B}_{i}}\right) \) and consider a random order \( \pi \) of \( X \) . For each \( i \) , \( 1 < i < k \), let \( {X}_{i} \) be the event that all the elements of \( {A}_{i} \) precede all those of \( {B}_{i} \) in this order. Clearly \( \Pr \l... | Yes |
Theorem 1.4.1 [Erdős (1965a)] Every set \( B = \left\{ {{b}_{1},\ldots ,{b}_{n}}\right\} \) of n nonzero integers contains a sum-free subset \( A \) of size \( \left| A\right| > \frac{1}{3}n \) . | Proof. Let \( p = {3k} + 2 \) be a prime, which satisfies \( p > 2\mathop{\max }\limits_{{1 \leq i \leq n}}\left| {b}_{i}\right| \) and put \( C = \) \( \{ k + 1, k + 2,\ldots ,{2k} + 1\} \) . Observe that \( C \) is a sum-free subset of the cyclic group \( {Z}_{p} \) and that \( \frac{\left| C\right| }{p - 1} = \frac{... | Yes |
Theorem 1.5.1 Let \( \mathcal{F} \) be a family of \( m = {2}^{\left( {\frac{1}{2} + \delta }\right) n} \) subsets of \( X = \{ 1,2,\ldots, n\} \) ,\n\nwhere \( \delta > 0 \) . Then\n\[ d\left( \mathcal{F}\right) < {m}^{2 - \frac{{\delta }^{2}}{2}}\text{.} \] | Proof. Suppose (1.1) is false and pick independently \( t \) members \( {A}_{1},{A}_{2},\ldots ,{A}_{t} \) of \( \mathcal{F} \) with repetitions at random, where \( t \) is a large positive integer, to be chosen later.\n\nWe will show that with positive probability \( \left| {{A}_{1} \cup {A}_{2} \cup \ldots \cup {A}_{... | Yes |
Lemma 1 For \( 0 \leq s \leq n - 1 \) set \( {A}_{s} = \{ s, s + 1,\ldots, s + k - 1\} \) where addition is modulo \( n \) . Then \( \mathcal{F} \) can contain at most \( k \) of the sets \( {A}_{s} \) . | Proof. Fix some \( {A}_{s} \in \mathcal{F} \) . All other sets \( {A}_{t} \) that intersect \( {A}_{s} \) can be partitioned into \( k - 1 \) pairs \( \left\{ {{A}_{s - i},{A}_{s + k - i}}\right\} ,\left( {1 \leq i \leq k - 1}\right) \), and the members of each such pair are disjoint. The result follows, since \( \math... | Yes |
Theorem 2.1.1 There is a tournament \( T \) with \( n \) players and at least \( n!{2}^{-\left( {n - 1}\right) } \) Hamiltonian Paths. | Proof. In the random tournament let \( X \) be the number of Hamiltonian paths. For each permutation \( \sigma \) let \( {X}_{\sigma } \) be the indicator random variable for \( \sigma \) giving a Hamiltonian path - i.e., satisfying \( \left( {\sigma \left( i\right) ,\sigma \left( {i + 1}\right) }\right) \in T \) for \... | Yes |
Theorem 2.2.1 Let \( G = \left( {V, E}\right) \) be a graph with \( n \) vertices and e edges. Then \( G \) contains a bipartite subgraph with at least \( e/2 \) edges. | Proof. Let \( T \subseteq V \) be a random subset given by \( \Pr \left\lbrack {x \in T}\right\rbrack = 1/2 \), these choices mutually independent. Set \( B = V - T \) . Call an edge \( \{ x, y\} \) crossing if exactly one of \( x, y \) are in \( T \) . Let \( X \) be the number of crossing edges. We decompose\n\n\[ \n... | Yes |
Theorem 2.2.2 If \( G \) has \( {2n} \) vertices and e edges then it contains a bipartite subgraph with at least \( \frac{en}{{2n} - 1} \) edges. If \( G \) has \( {2n} + 1 \) vertices and e edges then it contains a bipartite subgraph with at least \( \frac{e\left( {n + 1}\right) }{{2n} + 1} \) edges. | Proof. When \( G \) has \( {2n} \) vertices let \( T \) be chosen uniformly from among all \( n \) -element subsets of \( V \) . Any edge \( \{ x, y\} \) now has probability \( \frac{n}{{2n} - 1} \) of being crossing and the proof concludes as before. When \( G \) has \( {2n} + 1 \) vertices choose \( T \) uniformly fr... | No |
Lemma 2.2.4 Let \( {P}_{k} \) denote the set of all homogeneous polynomials \( f\left( {{p}_{1},\ldots ,{p}_{k}}\right) \) of degree \( k \) with all coefficients having absolute value at most one and \( {p}_{1}{p}_{2}\cdots {p}_{k} \) having coefficient one. Then for all \( f \in {P}_{k} \) there exist \( {p}_{1},\ldo... | Proof. Set\n\n\[ M\left( f\right) = \mathop{\max }\limits_{{{p}_{1},\ldots ,{p}_{k} \in \left\lbrack {0,1}\right\rbrack }}\left| {f\left( {{p}_{1},\ldots ,{p}_{k}}\right) }\right| . \]\n\nFor \( f \in {P}_{k}, M\left( f\right) > 0 \) as \( f \) is not the zero polynomial. As \( {P}_{k} \) is compact and \( M : {P}_{k} ... | Yes |
Theorem 2.3.1 There is a two-coloring of \( {K}_{n} \) with at most\n\n\[ \left( \begin{array}{l} n \\ a \end{array}\right) {2}^{1 - \left( \begin{array}{l} a \\ 2 \end{array}\right) } \]\n\nmonochromatic \( {K}_{a} \) . | Proof.[outline] Take a random coloring . Let \( X \) be the number of monochromatic \( {K}_{a} \) and find \( E\left\lbrack X\right\rbrack \) . For some coloring the value of \( X \) is at most this expectation. - | No |
Theorem 2.3.2 There is a two-coloring of \( {K}_{m, n} \) with at most\n\n\[ \left( \begin{matrix} m \\ a \end{matrix}\right) \left( \begin{array}{l} n \\ b \end{array}\right) {2}^{1 - {ab}} \]\n\nmonochromatic \( {K}_{a, b} \) . | Proof.[outline] Take a random coloring . Let \( X \) be the number of monochromatic \( {K}_{a, b} \) and find \( E\left\lbrack X\right\rbrack \) . For some coloring the value of \( X \) is at most this expectation. ∎ | No |
Theorem 2.4.1 Let \( {v}_{1},\ldots ,{v}_{n} \in {R}^{n} \), all \( \left| {v}_{i}\right| = 1 \) . Then there exist \( {\epsilon }_{1},\ldots ,{\epsilon }_{n} = \pm 1 \) so that\n\n\[ \left| {{\epsilon }_{1}{v}_{1} + \ldots + {\epsilon }_{n}{v}_{n}}\right| \leq \sqrt{n} \]\n\nand also there exist \( {\epsilon }_{1},\ld... | Proof. Let \( {\epsilon }_{1},\ldots ,{\epsilon }_{n} \) be selected uniformly and independently from \( \{ - 1, + 1\} \) . Set\n\n\[ X = {\left| {\epsilon }_{1}{v}_{1} + \ldots + {\epsilon }_{n}{v}_{n}\right| }^{2} \]\n\nThen\n\n\[ X = \mathop{\sum }\limits_{{i = 1}}^{n}\mathop{\sum }\limits_{{j = 1}}^{n}{\epsilon }_{... | Yes |
Theorem 2.4.2 Let \( {v}_{1},\ldots ,{v}_{n} \in {R}^{n} \), all \( \left| {v}_{i}\right| \leq 1 \) . Let \( {p}_{1},\ldots ,{p}_{n} \in \left\lbrack {0,1}\right\rbrack \) be arbitrary and set \( w = {p}_{1}{v}_{1} + \ldots + {p}_{n}{v}_{n} \) . Then there exist \( {\epsilon }_{1},\ldots ,{\epsilon }_{n} \in \{ 0,1\} \... | Proof. Pick \( {\epsilon }_{i} \) independently with\n\n\[ \Pr \left\lbrack {{\epsilon }_{i} = 1}\right\rbrack = {p}_{i},\;\Pr \left\lbrack {{\epsilon }_{i} = 0}\right\rbrack = 1 - {p}_{i} \]\n\nThe random choice of \( {\epsilon }_{i} \) gives a random \( v \) and a random variable\n\n\[ X = {\left| w - v\right| }^{2} ... | Yes |
Theorem 2.5.1 Let \( {a}_{ij} = \pm 1 \) for \( 1 \leq i, j \leq n \) . Then there exist \( {x}_{i},{y}_{j} = \pm 1 \) , \( 1 \leq i, j \leq n \) so that \[ \mathop{\sum }\limits_{{i = 1}}^{n}\mathop{\sum }\limits_{{j = 1}}^{n}{a}_{ij}{x}_{i}{y}_{j} \geq \left( {\sqrt{\frac{2}{\pi }} + o\left( 1\right) }\right) {n}^{3/... | Proof.[Theorem 2.5.1] Forget the \( x \) ’s. Let \( {y}_{1},\ldots ,{y}_{n} = \pm 1 \) be selected independently and uniformly and set \[ {R}_{i} = \mathop{\sum }\limits_{{j = 1}}^{n}{a}_{ij}{y}_{j} \] \[ R = \mathop{\sum }\limits_{{i = 1}}^{n}\left| {R}_{i}\right| \] Fix \( i \) . Regardless of \( {a}_{ij},{a}_{ij}{y}... | Yes |
Theorem 1 [Brégman's Theorem]\n\n\\[ \n\\operatorname{per}\\left( A\\right) \\leq \\mathop{\\prod }\\limits_{{1 \\leq i \\leq n}}{\\left( {r}_{i}!\\right) }^{1/{r}_{i}} \n\\] | ## Proof.\n\nWe show this for any fixed \\( \\tau \\) . Set \\( ▩ = 1 \\) for convenience of notation. We use induction on the size of the matrix. Reorder, for convenience, so that the first row has ones in the first \\( r \\) columns where \\( r = {r}_{1} \\) . For \\( 1 \\leq j \\leq r \\) let \\( {t}_{j} \\) be the ... | Yes |
Lemma 2\n\n\[ \n{\left( \mathop{\prod }\limits_{{j = 1}}^{r}{t}_{j}^{{t}_{j}}\right) }^{1/r} \geq {t}^{t} \n\] | Proof. Taking logarithms this is equivalent to\n\n\[ \n\frac{1}{r}\mathop{\sum }\limits_{{j = 1}}^{r}{t}_{j}\ln {t}_{j} \geq t\ln t \n\]\n\nwhich follows from the convexity of the function \( f\left( x\right) = x\ln x \) . ∎ | Yes |
Theorem 3.1.1 For any integer \( n \)\n\n\[ R\left( {k, k}\right) > n - \left( \begin{array}{l} n \\ k \end{array}\right) {2}^{1 - \left( \begin{array}{l} k \\ 2 \end{array}\right) } \] | Proof. Consider a random two-coloring of the edges of \( {K}_{n} \) obtained by coloring each edge independently either red or blue, where each color is equally likely. For any set \( R \) of \( k \) vertices let \( {X}_{R} \) be the indicator random variable for the event that the induced subgraph of \( {K}_{n} \) on ... | Yes |
Theorem 3.1.3 For all integers \( n \) and \( p \in \left\lbrack {0,1}\right\rbrack \)\n\n\[ R\left( {k, l}\right) > n - \left( \begin{array}{l} n \\ k \end{array}\right) {p}^{\left( \begin{array}{l} k \\ 2 \end{array}\right) } - \left( \begin{array}{l} n \\ l \end{array}\right) {\left( 1 - p\right) }^{\left( \begin{ar... | Proof. In both cases we consider a random two-coloring of \( {K}_{n} \) obtained by coloring each edge independently either red or blue, where each edge is red with probability \( p \) . Let \( X \) be the number of red \( k \) -sets plus the number of blue \( l \) -sets. Linearity of Expectation gives\n\n\[ E\left\lbr... | No |
Theorem 3.2.1 Let \( G = \left( {V, E}\right) \) have \( n \) vertices and \( {nd}/2 \) edges, \( d \geq 1 \) . Then \( \alpha \left( G\right) \geq n/{2d} \) . | Proof. Let \( S \subseteq V \) be a random subset defined by\n\n\[ \Pr \left\lbrack {v \in S}\right\rbrack = p \]\n\n\( p \) to be determined, the events \( v \in S \) being mutually independent. Let \( X = \left| S\right| \) and let \( Y \) be the number of edges in \( {\left. G\right| }_{S} \) . For each \( e = \{ i,... | Yes |
Theorem 3.3.1 There is a set \( S \) of \( n \) points in the unit square \( U \) such that \( T\left( S\right) \geq \) \( 1/\left( {{100}{n}^{2}}\right) \) . | ## Proof.\n\nWe first make a calculation. Let \( P, Q, R \) be independently and uniformly selected from \( U \) and let \( \mu = \mu \left( {PQR}\right) \) denote the area of the triangle \( {PQR} \) . We bound \( \Pr \left\lbrack {\mu \leq \epsilon }\right\rbrack \) as follows. Let \( x \) be the distance from \( P \... | Yes |
Theorem 3.4.1 Let \( C \) be bounded, convex, and centrally symmetric around the origin. Then\n\n\[ \delta \left( C\right) \geq {2}^{-d - 1} \] | Proof. Let \( P, Q \) be selected independently and uniformly from \( B\left( x\right) \) and consider the event \( \left( {C + P}\right) \cap \left( {C + Q}\right) \neq \varnothing \) . For this to occur we must have, for some \( {c}_{1},{c}_{2} \in C \)\n\n\[ P - Q = {c}_{1} - {c}_{2} = 2\frac{{c}_{1} - {c}_{2}}{2} \... | Yes |
Corollary 3.5.2 \( m\left( n\right) = \Omega \left( {{2}^{n}{\left( n/\ln n\right) }^{1/2}}\right) \) | Proof. Bound \( 1 - p \leq {e}^{-p} \) . The function \( k{e}^{-{pn}} + {k}^{2}p \) is minimized at \( p = \) \( \ln \left( {n/k}\right) /n \) . Substituting back in, if\n\n\[ \frac{{k}^{2}}{n}\left\lbrack {1 + \ln \left( {n/k}\right) }\right\rbrack < 1 \]\n\nthen the condition of Theorem 3.5.1 holds. This inequality i... | Yes |
Lemma 3.5.3 \( E\left\lbrack {{\left( 1 + p\right) }^{i}{\left( 1 - p\right) }^{j}}\right\rbrack \leq 1 \) . | Proof. Fix a matching between \( e - \{ v\} \) and \( f - \{ v\} \), think of Mr. &Mrs. Jones; Mr. & Mrs. Smith, etc. Condition on how many of each pair (two Joneses, one Smith, no Taylors,..) come before \( v \) . The conditional expectation of \( {\left( 1 + p\right) }^{i}{\left( 1 - p\right) }^{j} \) splits into fac... | Yes |
Corollary 3.6.2 Under the assumptions of the theorem there exists a packing \( P \) of size \( \sim N/\left( {k + 1}\right) \) . | Proof. We have defined a random process which gives a packing with expected size \( \sim N/\left( {k + 1}\right) \) and our usual magic implies such a \( P \) must exist. - | No |
Theorem 1 [Erdős (1959)] For all \( k, l \) there exists a graph \( G \) with \( \operatorname{girth}\left( G\right) > l \) and \( \chi \left( G\right) > k \) . | Proof. Fix \( \theta < 1/l \) and let \( G \sim G\left( {n, p}\right) \) with \( p = {n}^{\theta - 1} \) . (I.e., \( G \) is a random graph on \( n \) vertices chosen by picking each pair of vertices as an edge randomly and independently with probability \( p \) ). Let \( X \) be the number of cycles of size at most \(... | Yes |
Theorem 4.1.1 [Chebyschev’s Inequality] For any positive \( \lambda \)\n\n\[ \Pr \left\lbrack {\left| {X - \mu }\right| \geq {\lambda \sigma }}\right\rbrack \leq \frac{1}{{\lambda }^{2}} \] | Proof.\n\n\[ {\sigma }^{2} = \operatorname{Var}\left\lbrack X\right\rbrack = E\left\lbrack {\left( X - \mu \right) }^{2}\right\rbrack \geq {\lambda }^{2}{\sigma }^{2}\Pr \left\lbrack {\left| {X - \mu }\right| \geq {\lambda \sigma }}\right\rbrack . \] | Yes |
Theorem 4.3.1\n\n\[ \Pr \left\lbrack {X = 0}\right\rbrack \leq \frac{\operatorname{Var}\left\lbrack X\right\rbrack }{E{\left\lbrack X\right\rbrack }^{2}} \] | Proof. Set \( \lambda = \mu /\sigma \) in Chebyschev’s Inequality. Then\n\n\[ \Pr \left\lbrack {X = 0}\right\rbrack \leq \Pr \left\lbrack {\left| {X - \mu }\right| \geq {\lambda \sigma }}\right\rbrack \leq \frac{1}{{\lambda }^{2}} = \frac{{\sigma }^{2}}{{\mu }^{2}}. \] | Yes |
Corollary 4.3.3 If \( \operatorname{Var}\left\lbrack X\right\rbrack = o\left( {E{\left\lbrack X\right\rbrack }^{2}}\right) \) then \( X \sim E\left\lbrack X\right\rbrack \) a.a. | Suppose again \( X = {X}_{1} + \ldots + {X}_{m} \) where \( {X}_{i} \) is the indicator random variable for event \( {A}_{i} \). For indices \( i, j \) write \( i \sim j \) if \( i \neq j \) and the events \( {A}_{i},{A}_{j} \) are not independent. We set (the sum over ordered pairs)\n\n\[ \Delta = \mathop{\sum }\limit... | Yes |
Theorem 4.4.1 The property \( \omega \left( G\right) \geq 4 \) has threshold function \( {n}^{-2/3} \) . | Proof. For every 4-set \( S \) of vertices in \( G\left( {n, p}\right) \) let \( {A}_{S} \) be the event \ | No |
Theorem 4.4.2 Let \( H \) be a balanced graph with \( v \) vertices and e edges. Let \( A\left( G\right) \) be the event that \( H \) is a subgraph (not necessarily induced) of \( G \) . Then \( p = {n}^{-v/e} \) is the threshold function for \( A \) . | Proof. We follow the argument of Theorem 4.4.1. For each \( v \) -set \( S \) let \( {A}_{S} \) be the event that \( {\left. G\right| }_{S} \) contains \( H \) as a subgraph. Then\n\n\[ \n{p}^{e} \leq \Pr \left\lbrack {A}_{S}\right\rbrack \leq v!{p}^{e} \n\]\n\n(Any particular placement of \( H \) has probability \( {p... | Yes |
Theorem 4.4.3 In the notation of Theorem 4.4.2 if \( H \) is not balanced then \( p = {n}^{-v/e} \) is not the threshold function for \( A \) . | Proof. Let \( {H}_{1} \) be a subgraph of \( H \) with \( {v}_{1} \) vertices, \( {e}_{1} \) edges and \( {e}_{1}/{v}_{1} > e/v \) . Let \( \alpha \) satisfy \( v/e < \alpha < {v}_{1}/{e}_{1} \) and set \( p = {n}^{-\alpha } \) . The expected number of copies of \( {H}_{1} \) is then \( o\left( 1\right) \) so almost al... | Yes |
Theorem 4.4.4 Let \( H \) be strictly balanced with \( v \) vertices, \( e \) edges and a automor-phisms. Let \( X \) be the number of copies of \( H \) in \( G\left( {n, p}\right) \) . Assume \( p > > {n}^{-v/e} \) . Then almost always\n\n\[X \sim \frac{{n}^{v}{p}^{e}}{a}\] | Proof. Label the vertices of \( H \) by \( 1,\ldots, v \) . For each ordered \( {x}_{1},\ldots ,{x}_{v} \) let \( {A}_{{x}_{1},\ldots ,{x}_{v}} \) be the event that \( {x}_{1},\ldots ,{x}_{v} \) provides a copy of \( H \) in that order. Specifically we define\n\n\[{A}_{{x}_{1},\ldots ,{x}_{v}} : \{ i, j\} \in E\left( H... | Yes |
Theorem 4.4.5 Let \( H \) be any fixed graph. For every subgraph \( {H}^{\prime } \) of \( H \) (including \( H \) itself) let \( {X}_{{H}^{\prime }} \) denote the number of copies of \( {H}^{\prime } \) in \( G\left( {n, p}\right) \) . Assume \( p \) is such that \( E\left\lbrack {X}_{{H}^{\prime }}\right\rbrack \righ... | Proof. Let \( H \) have \( v \) vertices and \( e \) edges. As in Theorem 4.4.4 it suffices to show \( {\Delta }^{ * } = o\left( {E\left\lbrack X\right\rbrack }\right) \) . We split \( {\Delta }^{ * } \) into a finite number of terms. For each \( {H}^{\prime } \) with \( w \) vertices and \( f \) edges we have those \(... | Yes |
Theorem 4.5.1 Let \( k = k\left( n\right) \) satisfy \( k \sim 2{\log }_{2}n \) and \( f\left( k\right) \rightarrow \infty \) . Then almost always \( \omega \left( G\right) \geq k \) . | Proof. For each \( k \) -set \( S \) let \( {A}_{S} \) be the event \ | No |
Theorem 4.7.1 For every integer \( r \geq 2 \) and reals \( k \geq 1 \) and \( a > 0 \), there are \( \gamma = \gamma \left( {r, k, a}\right) > 0 \) and \( {d}_{0} = {d}_{0}\left( {r, k, a}\right) \) such that for every \( n \geq D \geq {d}_{0} \) the following holds.\n\nEvery \( r \) -uniform hypergraph \( H = \left( ... | The basic idea in the proof is simple. Fixing a small \( \epsilon > 0 \) one shows that a random set of roughly \( {\epsilon n}/r \) edges has, with high probability, only some \( O\left( {{\epsilon }^{2}n}\right) \) vertices covered more than once, and hence covers at least \( {\epsilon n} - O\left( {{\epsilon }^{2}n}... | Yes |
Lemma 4.7.2 For every integer \( r \geq 2 \) and reals \( K \geq 1 \) and \( \epsilon > 0 \), and for every real \( {\delta }^{\prime } > 0 \), there are \( \delta = \delta \left( {r, K,\epsilon ,{\delta }^{\prime }}\right) > 0 \) and \( {D}_{0} = {D}_{0}\left( {r, K,\epsilon ,{\delta }^{\prime }}\right) \) such that f... | Proof. Throughout the proof we assume, whenever this is needed, that \( D \) (and hence \( n \) ) are sufficiently large. We denote by \( {\delta }_{1},{\delta }_{2},\cdots \) positive constants (that can be explicitly estimated) that tend to 0 when \( \delta \) tends to 0 and \( D \) tends to infinity (for fixed \( r,... | Yes |
Theorem 4.7.3 (Rödl) For \( k, l \) fixed,\n\n\[ M\left( {n, k, l}\right) \leq \left( {1 + o\left( 1\right) }\right) \frac{\left( \begin{matrix} n \\ l \end{matrix}\right) }{\left( \begin{matrix} k \\ l \end{matrix}\right) } \]\n\nwhere the \( o\left( 1\right) \) term tends to zero as \( n \) tends to infinity. | Proof. Put \( r = \left( \begin{array}{l} k \\ l \end{array}\right) \) and let \( H \) be the \( r \) -uniform hypergraph whose vertices are all \( l \) -subsets of \( \{ 1,2,\ldots, n\} \), and whose edges are all collections of \( \left( \begin{array}{l} k \\ l \end{array}\right) l \) -tuples that lie in a \( k \) -s... | Yes |
There exists a positive constant \( c \) such that for every \( n \)\n\n\[ P\left( n\right) \leq c{n}^{3/2}\frac{n!}{{2}^{n - 1}} \] | For a tournament \( T \), denote by \( P\left( T\right) \) the number of directed Hamilton paths of \( T \) . Similarly, \( C\left( T\right) \) denotes the number of directed Hamilton cycles of \( T \), and \( F\left( T\right) \) denotes the number of spanning subgraphs of \( T \) in which the indegree and the outdegre... | Yes |
For every two integers \( a, b \) satisfying \( b \geq a + 2 > a \geq 1 \) the inequality\n\n\[{\left( a!\right) }^{1/a} \cdot {\left( b!\right) }^{1/b} < {\left( \left( a + 1\right) !\right) }^{1/\left( {a + 1}\right) } \cdot {\left( \left( b - 1\right) !\right) }^{1/\left( {b - 1}\right) }\]\n\nholds. | Proof. The assertion is simply that \( f\left( a\right) < f\left( {b - 1}\right) \), for the function \( f \) defined by \( f\left( a\right) = {\left( a!\right) }^{1/a}/{\left( \left( a + 1\right) !\right) }^{1/\left( {a + 1}\right) } \) . Thus, it suffices to show that for every integer \( x \geq 2, f\left( {x - 1}\ri... | Yes |
Corollary 3 Define \( g\left( x\right) = {\left( x!\right) }^{1/x} \) . For every integer \( S \geq n \) the maximum of the function \( \mathop{\prod }\limits_{{i = 1}}^{n}g\left( {x}_{i}\right) \) subject to the constraints \( \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i} = S \) and \( {x}_{i} \geq 1 \) are integers, is ... | Proof. If there are two indices \( i \) and \( j \) such that \( {x}_{i} \geq {x}_{j} + 2 \) then, by Lemma 2, the value of the product would increase once we add one to \( {\mathbf{x}}_{\mathbf{j}} \) and subtract one from \( {\mathbf{x}}_{\mathbf{i}} \) . - | No |
Proposition 4 For every tournament \( T \) on \( n \) vertices\n\n\[ C\\left( T\\right) \\leq F\\left( T\\right) \\leq \\left( {1 + o\\left( 1\\right) }\\right) \\frac{\\sqrt{\\pi }}{\\sqrt{2}e}{n}^{3/2}\\frac{\\left( {n - 1}\\right) !}{{2}^{n}}. \] | To complete the proof of the theorem, we have to derive a bound for the number of Hamilton paths in a tournament from the above result. Given a tournament \( S \) on \( n \) vertices, let \( T \) be the random tournament obtained from \( S \) by adding to it a new vertex \( y \) and by orienting each edge connecting \(... | Yes |
Corollary 5.1.2 [The Local Lemma; Symmetric Case] Let \( {A}_{1},{A}_{2},\ldots ,{A}_{n} \) be events in an arbitrary probability space. Suppose that each event \( {A}_{i} \) is mutually independent of a set of all the other events \( {A}_{j} \) but at most \( d \), and that \( \Pr \left( {A}_{i}\right) \leq p \) for a... | Proof. If \( d = 0 \) the result is trivial. Otherwise, by the assumption there is a dependency digraph \( D = \left( {V, E}\right) \) for the events \( {A}_{1},\ldots ,{A}_{n} \) in which for each \( i, \mid \{ j \) : \( \left( {i, j}\right) \in E\} \mid \leq d \) . The result now follows from Lemma 5.1.1 by taking \(... | Yes |
Theorem 5.2.1 Let \( H = \left( {V, E}\right) \) be a hypergraph in which every edge has at least \( k \) elements, and suppose that each edge of \( H \) intersects at most \( d \) other edges. If \( e\left( {d + 1}\right) \leq {2}^{k - 1} \) then \( H \) has property \( B \) . | Proof. Color each vertex \( v \) of \( H \), randomly and independently, either blue or red (with equal probability). For each edge \( f \in E \), let \( {A}_{f} \) be the event that \( f \) is monochromatic. Clearly \( \Pr \left( {A}_{f}\right) = 2/{2}^{\left| f\right| } \leq 1/{2}^{k - 1} \) . Moreover, each event \(... | Yes |
Theorem 5.2.2 Let \( m \) and \( k \) be two positive integers satisfying\n\n\[ e\left( {m\left( {m - 1}\right) + 1}\right) k{\left( 1 - \frac{1}{k}\right) }^{m} \leq 1.\]\n\n(5.7)\n\nThen, for any set \( S \) of \( m \) real numbers there is a \( k \) -coloring so that each translation \( x + S \) (for \( x \in \mathb... | Proof. We first fix a finite subset \( X \subseteq \mathbb{R} \) and show the existence of a \( k \) -coloring so that each translation \( x + S \) (for \( x \in X \) ) is multicolored. This is an easy consequence of the Local Lemma. Indeed, put \( Y = \mathop{\bigcup }\limits_{{x \in X}}\left( {x + S}\right) \) and le... | Yes |
Note that it is impossible, in general, to apply the Local Lemma to an infinite number of events and conclude that in some point of the probability space none of them holds. In fact, there are trivial examples of countably many mutually independent events \( {A}_{i} \), satisfying \( \Pr \left( {A}_{i}\right) = 1/2 \) ... | Thus the compactness argument is essential in the above proof. | No |
Proposition 5.3.1 If \( e\left( {\left( \begin{array}{l} k \\ 2 \end{array}\right) \left( \begin{matrix} n \\ k - 2 \end{matrix}\right) + 1}\right) \cdot {2}^{1 - \left( \begin{array}{l} k \\ 2 \end{array}\right) } < 1 \) then \( R\left( {k, k}\right) > n \) . | A short computation shows that this gives \( R\left( {k, k}\right) > \frac{\sqrt{2}}{e}\left( {1 + o\left( 1\right) }\right) k{2}^{k/2} \) , only a factor 2 improvement on the bound obtained by the straightforward probabilistic method. | Yes |
Theorem 5.4.1 Let \( \mathcal{F} = {\left\{ {B}_{i}\right\} }_{i \in I} \) be a \( k \) -fold covering of the 3 dimensional Euclidean space by open unit balls. Suppose, further, than no point of \( {\mathbf{R}}^{3} \) is contained in more than \( t \) members of \( \mathcal{F} \) . If\n\n\[ e \cdot {t}^{3}{2}^{18}/{2}^... | Proof. Define an infinite hypergraph \( H = \left( {V\left( H\right), E\left( H\right) }\right) \) as follows. The set of vertices of \( H, V\left( H\right) \), is simply \( \mathcal{F} = {\left\{ {B}_{i}\right\} }_{i \in I} \) . For each \( x \in {\mathbb{R}}^{3} \) let \( {E}_{x} \) be the set of balls \( {B}_{i} \in... | Yes |
Proposition 5.5.3 Let \( H = \left( {V, E}\right) \) be a graph with maximum degree \( d \), and let \( V = {V}_{1} \cup {V}_{2} \cup \cdots \cup {V}_{r} \) be a partition of \( V \) into \( r \) pairwise disjoint sets. Suppose each set \( {V}_{i} \) is of cardinality \( \left| {V}_{i}\right| \geq {2ed} \), where \( e ... | Proof. Clearly we may assume that each set \( {V}_{i} \) is of cardinality precisely \( g = \lceil {2ed}\rceil \) (otherwise, simply replace each \( {V}_{i} \) by a subset of cardinality \( g \) of it, and replace \( H \) by its induced subgraph on the union of these \( r \) new sets). Let us pick from each set \( {V}_... | Yes |
Theorem 5.5.4 Let \( G = \\left( {U, F}\\right) \) be a d-regular digraph with directed girth \( g \\geq 8 \) ed. Then\n\n\[ \n\\operatorname{dla}\\left( G\\right) = d + 1 \n\] | Proof. As is well known, \( F \) can be partitioned into \( d \) pairwise disjoint 1-regular spanning subgraphs \( {F}_{1},\\ldots ,{F}_{d} \) of \( G \) . (This is an easy consequence of the Hall-König Theorem; let \( H \) be the bipartite graph whose two classes of vertices \( A \) and \( B \) are copies of \( U \), ... | Yes |
Lemma 5.5.5 Let \( G = \left( {V, E}\right) \) be a \( d \) -regular directed graph, where \( d \) is sufficiently large, and let \( p \) be an integer satisfying \( {10}\sqrt{d} \leq p \leq {20}\sqrt{d} \) . Then, there is a \( p \) - coloring of the vertices of \( G \) by the colors \( 0,1,2,\ldots, p - 1 \) with the... | Proof. Let \( f : V \rightarrow \{ 0,1,\ldots, p - 1\} \) be a random vertex coloring of \( V \) by \( p \) colors, where for each \( v \in V, f\left( v\right) \in \{ 0,1,\ldots, p - 1\} \) is chosen according to a uniform distribution. For every vertex \( v \in V \) and every color \( i,0 \leq i < p \), let \( {A}_{v,... | Yes |
Theorem 5.5.6 There is an absolute constant \( c > 0 \) such that for every \( d \) -regular digrapn \( G \)\n\n\[ \mathrm{d}\operatorname{la}\left( G\right) \leq d + c{d}^{3/4}{\left( \log d\right) }^{1/2}. \] | We note that by being a little more careful, we can improve the error term to \( {c}^{\prime }{d}^{2/3}{\left( \log d\right) }^{1/3} \) . Since the edges of any undirected \( d = {2f} \) -regular graph can be oriented so that the resulting digraph is \( f \) -regular, and since any \( \left( {{2f} - 1}\right) \) -regul... | No |
Theorem 5.7.1 Let \( n, d \) be such that, setting \( D = d{\left( d - 1\right) }^{3} \) there exists a decomposition \( n = {n}_{1} + {n}_{2} + {n}_{3} \) with\n\n\[ \n{16D}\left( {1 + d}\right) < {2}^{{n}_{1}} \]\n\n\[ \n{16D}\left( {1 + d}\right) < {2}^{{n}_{2}} \]\n\n\[ \n{2e}\left( {1 + d}\right) < {2}^{{n}_{3}}. ... | Proof. The First Pass. During this pass points will be either Red, Blue, uncolored or saved. We move through the points \( j \in \Omega \) sequentially, coloring them Red or Blue at random, flipping a fair coin. After each \( j \) is colored we check all \( {A}_{i} \ni j \) . If \( {A}_{i} \) now has \( {n}_{1} \) poin... | Yes |
Theorem 1 [Alon and Linial (1989) ] If \( e\left( {{\Delta \delta } + 1}\right) {\left( 1 - \frac{1}{k}\right) }^{\delta } < 1 \) then \( D \) contains a (directed, simple) cycle of length \( 0\left( {\;\operatorname{mod}\;k}\right) \) . | Proof. Clearly we may assume that every outdegree is precisely \( \delta \), since otherwise we can consider a subgraph of \( D \) with this property.\n\nLet \( f : V \rightarrow \{ 0,1,\ldots, k - 1\} \) be a random coloring of \( V \), obtained by choosing, for each \( v \in V, f\left( v\right) \in \{ 0,\ldots, k - 1... | Yes |
Corollary 6.1.4 Let \( X \) be a family of subsets of a finite set \( N \) and define\n\n\[ \nX \smallsetminus X = \left\{ {F \smallsetminus {F}^{\prime } : F,{F}^{\prime } \in X}\right\} .\n\]\n\nThen \( \;\left| {X \smallsetminus X}\right| \geq \left| X\right| \) . | ## Proof.\n\nLet \( L \) be the distributive lattice of all subsets of \( N \) . By applying Corollary 6.1.3 to \( X \) and \( Y = \{ N \smallsetminus F : F \in X\} \) we obtain:\n\n\[ \n{\left| X\right| }^{2} = \left| X\right| \cdot \left| Y\right| \leq \left| {X \cup Y}\right| \cdot \left| {X \cap Y}\right| = {\left|... | Yes |
Theorem 6.2.1 [The \( {FKG} \) inequality]\n\nLet \( L \) be a finite distributive lattice and let \( \mu : L \rightarrow {\mathbb{R}}^{ + } \) be a log-supermodular function. Then, for any two increasing functions \( f, g : L \rightarrow {\mathbb{R}}^{ + } \) we have\n\n\[ \left( {\mathop{\sum }\limits_{{x \in L}}\mu ... | Proof.\n\nDefine four functions \( \alpha ,\beta ,\gamma ,\delta : L \rightarrow {\mathbb{R}}^{ + } \) as follows. For each \( x \in L \)\n\n\[ \alpha \left( x\right) \; = \mu \left( x\right) f\left( x\right) ,\;\beta \left( x\right) = \mu \left( x\right) g\left( x\right) \]\n\n\[ \gamma \left( x\right) = \mu \left( x\... | Yes |
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