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Theorem 4. Let \( G \) be a graph of order \( n \) in which every cycle has length at most \( k\left( {k \geq 2}\right) \) . Then\n\n\[ e\left( G\right) \leq \frac{k}{2}\left( {n - 1}\right) \]\n\nA graph is an extremal graph iff it is connected and all its blocks are complete graphs of order \( k \) . | The proof of this result is somewhat more involved than that of Theorem 3. Since a convenient way of presenting it uses 'simple transforms' to be introduced in \( §3 \), the proof is left as an exercise (Exercise 25) with a detailed hint. | No |
Theorem 6. \( \operatorname{ex}\left( {n;{K}^{r}}\right) = {t}_{r - 1}\left( n\right) \) and \( {T}_{r - 1}\left( n\right) \) is the unique graph of order \( n \) and size \( {t}_{r - 1}\left( n\right) \) that does not contain a complete graph of order \( r \) . | 1st Proof. Since \( {T}_{r - 1}\left( n\right) \) is the unique \( \left( {r - 1}\right) \) -partite graph of order \( n \) and maximum size, both assertions follow from Theorem 5. | No |
Lemma 7. Let \( m, n, s, t, r, k \) be integers, \( 2 \leq s \leq m,2 \leq t \leq n,0 \leq k,0 \leq \) \( r < m \), and let \( G = {G}_{2}\left( {m, n}\right) \) be a graph of size \( z = {my} = {km} + r \) without a \( K\left( {s, t}\right) \) subgraph having \( s \) vertices in the first class and \( t \) in the seco... | Proof. Denote by \( {V}_{1} \) and \( {V}_{2} \) the vertex classes of \( G \) . We shall say that a \( t \) -set (i.e., a set with \( t \) elements) \( T \) of \( {V}_{2} \) belongs to a vertex \( x \in {V}_{1} \) if \( x \) is joined to every vertex in \( T \) . The number of \( t \) -sets belonging to a vertex \( x ... | Yes |
Theorem 8. \( z\left( {m, n;s, t}\right) \leq {\left( s - 1\right) }^{1/t}\left( {n - t + 1}\right) {m}^{1 - \left( {1/t}\right) } + \left( {t - 1}\right) m \) . | Proof. Let \( G = {G}_{2}\left( {m, n}\right) \) be an extremal graph for the function \( z\left( {m, n;s, t}\right) \) , that is let \( G \) be a bipartite graph of size \( z\left( {m, n;s, t}\right) = {my} \) without a \( K\left( {s, t}\right) \) subgraph. As \( y \leq n \), inequality (1) implies\n\n\[ \n{\left( y -... | Yes |
Theorem 9. \( \operatorname{ex}\left( {n;{K}_{2}\left( t\right) }\right) \leq \frac{1}{2}{\left( t - 1\right) }^{1/t}\left( {n - t + 1}\right) {n}^{1 - \left( {1/t}\right) } + \frac{1}{2}\left( {t - 1}\right) n \) | Proof. Let \( G \) be an extremal graph. As in Lemma 7, let us say that a \( t \) -set of the vertices belongs to a vertex \( x \) if \( x \) is joined to every vertex of the \( t \) -set. Since \( G \) does not contain a \( {K}_{2}\left( t\right) \), every \( t \) -set belongs to at most \( t - 1 \) vertices. Therefor... | Yes |
Theorem 10. \( z\left( {n, n;2,2}\right) \leq \frac{1}{2}n\left\{ {1 + {\left( 4n - 3\right) }^{1/2}}\right\} \) and equality holds for infinitely many values of \( n \) . | Proof. Note that with \( z = \frac{1}{2}n\left\{ {1 + {\left( 4n - 3\right) }^{1/2}}\right\} \) we have\n\n\[ \left( {z - n}\right) z = {n}^{2}\left( {n - 1}\right) .\n\]\n\nSuppose that there is a bipartite graph \( G = {G}_{2}\left( {n, n}\right) \) of size greater than \( z \) that does not contain a quadrilateral (... | Yes |
Lemma 12. Let \( k, n \) and \( t \) be natural numbers, \( t < n \), and let \( G \) be a graph with vertex set \( V\left( G\right) = \left\{ {{x}_{1},{x}_{2},\ldots ,{x}_{n}}\right\} \), whose \( k \) -closure \( {C}_{k}\left( G\right) \) contains at most \( t - 1 \) vertices of degree \( n - 1 \) . Then there are in... | Proof. The graph \( H = {C}_{k}\left( G\right) \) is not complete so we can define two indices \( i \) and \( j \) as follows:\n\n\[ j = \max \left\{ {l : {d}_{H}\left( {x}_{L}\right) \neq n - 1}\right\} \]\n\n\[ i = \max \left\{ {l : {x}_{l}{x}_{j} \notin E\left( H\right) }\right\} \]\n\nThen \( {x}_{i}{x}_{j} \notin ... | Yes |
Theorem 15. The graph \( G \) has no \( L \) -R edges. | Proof. Recall that there is no \( L - \left( {V\left( G\right) - V\left( S\right) }\right) \) edge, since \( S \) is a longest \( {x}_{0} \) -path, so in particular \( V\left( S\right) = V\left( P\right) \) for every transform \( P \) of \( S \) . Suppose \( {x}_{i}{x}_{j} \in E\left( G\right) \), where \( {x}_{i} \in ... | Yes |
Theorem 16. Let \( W \) be the set of vertices of even degree in a graph \( G \) and let \( {x}_{0} \) be a vertex of \( G \) . Then there are an even number of longest \( {x}_{0} \) -paths ending in \( W \) . | Proof. Let \( H \) be the graph whose vertex set is the set \( \sum \) of longest \( {x}_{0} \) -paths in \( G \), in which \( {P}_{1} \in \sum \) is joined to \( {P}_{2} \in \sum \) iff \( {P}_{2} \) is a simple transform of \( {P}_{1} \) . Since the degree of \( \bar{P} = {x}_{0}{x}_{1}\ldots {x}_{k} \in \sum \) in \... | Yes |
Theorem 17. Let \( G \) be a graph in which every vertex has odd degree. Then every edge of \( G \) is contained in an even number of Hamilton cycles. | Proof. Let \( {x}_{0}y \in E\left( G\right) \) . Then in \( {G}^{\prime } = G - {x}_{0}y \) only \( {x}_{0} \) and \( y \) have even degree, so in \( {G}^{\prime } \) there are an even number of longest \( {x}_{0} \) -paths that end in \( y \) . Thus either \( G \) has no Hamilton cycle that contains \( {x}_{0}y \) or ... | Yes |
Lemma 18. Let \( 0 < a < a + \varepsilon < 1 \) and put \( \eta = {\left( \varepsilon /2\right) }^{1/2} \) . Then every graph \( G \) of order \( n \geq {2a}/\varepsilon \) and size at least \( \frac{1}{2}\left( {a + \varepsilon }\right) {n}^{2} \) contains a subgraph \( H \) with \( \left| H\right| = h \geq {\eta n} \... | Proof. Define a sequence of graphs \( {G}_{0} = G \supset {G}_{1} \supset {G}_{2} \supset \ldots \) with \( \left| {G}_{k}\right| = \) \( n - k \) as follows. If \( {G}_{k} \) has a vertex \( {x}_{k} \) of degree less than \( a\left( {n - k}\right) \) then put \( {G}_{k + 1} = {G}_{k} - {x}_{k} \), if \( \delta \left( ... | Yes |
Lemma 19. Let \( r \geq 2,1 \leq t \leq q \) and \( N = n - \left( {r - 1}\right) q \geq 1 \) . Let \( G \) be a graph of order \( n \) that contains a \( {K}_{r - 1}\left( q\right) \), say \( \widetilde{K} \), but does not contain a \( {K}_{r}\left( t\right) \) . Then \( G \) has at most\n\n\[ e = \left( {\left( {r - ... | Proof. Let \( {S}_{1},{S}_{2},\ldots ,{S}_{r - 1} \) be the vertex classes of \( \widetilde{K} \) and put \( S = \mathop{\bigcup }\limits_{1}^{{r - 1}}{S}_{i} \) . Thus \( \left| {S}_{i}\right| = q \) and \( \left| S\right| = \left( {r - 1}\right) q \) . Let \( {x}_{1},{x}_{2},\ldots ,{x}_{N} \) be the vertices of \( G... | Yes |
Corollary 21. Let \( F = {K}_{r}\left( t\right) \), where \( r \geq 2 \) and \( t \geq 1 \) . Then the maximal size of a graph of order \( n \) without a \( {K}_{r}\left( t\right) \) is | \[ \operatorname{ex}\left( {n;F}\right) = \frac{1}{2}\left\{ {\frac{r - 2}{r - 1} + o\left( 1\right) }\right\} {n}^{2}. \] | Yes |
Let \( {F}_{1},{F}_{2},\ldots ,{F}_{l} \) be non-empty graphs. Denote by \( r \) the minimum of the chromatic numbers of the \( {F}_{i} \), that is let \( r \) be the minimum number for which at least one of the \( {F}_{i} \) is contained in an \( F = {K}_{r}\left( t\right) \) for some \( t \) . Then the maximal size o... | Proof. The Turán graph \( {T}_{r - 1}\left( n\right) \) does not contain any of the \( {F}_{i} \), so\n\n\[ \operatorname{ex}\left( {n;{F}_{1},{F}_{2},\ldots ,{F}_{l}}\right) \geq e\left( {{T}_{r - 1}\left( n\right) }\right) = {t}_{r - 1}\left( n\right) = \frac{1}{2}\left\{ {\frac{r - 2}{r - 1} + o\left( 1\right) }\rig... | Yes |
Corollary 23. The upper density of an infinite graph \( G \) is \( 1,\frac{1}{2},\frac{2}{3},\frac{3}{4},\ldots \), or 0 . Each of these values is the upper density of some infinite graph. | Proof. Let \( {G}_{r} \) be the complete \( r \) -partite graph with infinitely many vertices in each class. Since the density of \( {K}_{r}\left( t\right) \) tends to \( 1 - \left( {1/r}\right) \) as \( t \) tends to \( \infty \) , the upper density of \( {G}_{r} \) is \( 1 - \left( {1/r}\right) \), proving the second... | Yes |
Theorem 1. Let \( k = \mathop{\max }\limits_{H}\delta \left( H\right) \), where the maximum is taken over all spanned subgraphs of \( G \) . Then \( \chi \left( G\right) \leq k + 1 \) . | Proof. Write \( {x}_{n} \) for a vertex of degree at most \( k \), and put \( {H}_{n - 1} = G - \left\{ {x}_{n}\right\} \) . By assumption \( {H}_{n - 1} \) has a vertex of degree at most \( k \) . Let \( {x}_{n - 1} \) be one of them and put \( {H}_{n - 2} = {H}_{n - 1} - \left\{ {x}_{n - 1}\right\} = G - \left\{ {{x}... | Yes |
Theorem 3. Let \( G \) be a connected graph with maximal degree \( \Delta \) . Suppose \( G \) is neither a complete graph nor an odd cycle. Then \( \chi \left( G\right) \leq \Delta \) . | Proof. We know already that we may assume without loss of generality that \( G \) is 2-connected and \( \Delta \) -regular. Furthermore, we may assume that \( \Delta \geq 3 \) , since a 2-regular 3-chromatic graph is an odd cycle.\n\nIf \( G \) is 3-connected, let \( {x}_{n} \) be any vertex of \( G \) and let \( {x}_{... | Yes |
Theorem 4. Let \( H \) be a graph with \( n \geq 1 \) vertices, \( m \) edges and \( k \) components. Then\n\n\[ \n{p}_{H}\left( x\right) = \mathop{\sum }\limits_{{i = 0}}^{{n - k}}{\left( -1\right) }^{i}{a}_{i}{x}^{n - i} \n\]\n\nwhere \( {a}_{0} = 1,{a}_{1} = m \) and \( {a}_{i} > 0 \) for every \( i,0 \leq i \leq n ... | Proof. We apply induction on \( n + m \) . For \( n + m = 1 \) the assertions are trivial so we pass to the induction step. If \( m = 0 \), we are again home since in this case \( k = n \) and as every map \( f : V\left( H\right) \rightarrow \{ 1,2,\ldots, x\} \) is a colouring of \( H \), we have \( {p}_{H}\left( x\ri... | Yes |
Theorem 6. Every plane graph is 5-colourable. | Proof. Suppose the assertion is false and let \( G \) be a 6-chromatic plane graph with minimal number of vertices. By Theorem 12 of Chapter I \( G \) has a vertex \( x \) of degree at most 5 . Put \( H = G - x \) . Then \( H \) is 5-colourable, say with colours \( 1,2,\ldots ,5 \) . Each of these colours must be used ... | Yes |
Theorem 7. The number of colours needed to colour a graph drawn on an orientable surface of genus \( \gamma \geq 1 \) is at most\n\n\[ H\left( \gamma \right) = \left\lfloor {\frac{1}{2}\{ 7 + \sqrt{1 + {48\gamma }}\} }\right\rfloor \] | Proof. Let \( G \) be the graph in question and let \( H \) be any subgraph of \( G \) . By Theorem 1 we are home if we show that \( \delta \left( H\right) \leq H\left( \gamma \right) - 1 \) . If \( H \) has \( n \) vertices, it has at most \( {3n} + 6\left( {\gamma - 1}\right) \) edges so\n\n\[ \delta \left( H\right) ... | Yes |
Theorem 1. If \( s > 2 \) and \( t > 2 \) then\n\n\[ R\left( {s, t}\right) \leq R\left( {s - 1, t}\right) + R\left( {s, t - 1}\right) \] | Proof. (i) When proving (1) we may assume that \( R\left( {s - 1, t}\right) \) and \( R\left( {s, t - 1}\right) \) are finite. Let \( n = R\left( {s - 1, t}\right) + R\left( {s, t - 1}\right) \) and consider a colouring of the edges of \( {K}^{n} \) with red and blue. Wd have to show that this colouring contains either... | Yes |
Theorem 2. Let \( 1 < r < \min \{ s, t\} \) . Then \( {R}^{\left( r\right) }\left( {s, t}\right) \) is finite and \[ {R}^{\left( r\right) }\left( {s, t}\right) \leq {R}^{\left( r - 1\right) }\left( {{R}^{\left( r\right) }\left( {s - 1, t}\right) ,{R}^{\left( r\right) }\left( {s, t - 1}\right) }\right) + 1. \] | Proof. Both assertions follow if we prove the inequality under the assumption that \( {R}^{\left( r - 1\right) }\left( {u, v}\right) \) is finite for all \( u, v \), and \( {R}^{\left( r\right) }\left( {s - 1, t}\right) ,{R}^{\left( r\right) }\left( {s, t - 1}\right) \) are also finite.\n\nLet \( X \) be a set with \( ... | Yes |
Theorem 3. Let \( c : {A}^{\left( r\right) } \rightarrow \{ 1,\ldots, k\} \) be a \( k \) -colouring of the \( r \) -tuples \( \left( {1 \leq r < \infty }\right) \) of an infinite set \( A \) . Then \( A \) contains a monochromatic infinite set. | Proof. We apply induction on \( r \) . Note that the result is trivial for \( r = 1 \) so assume that \( r > 1 \) and the theorem holds for smaller values of \( r \) .\n\nPut \( {A}_{0} = A \) and pick an element \( {x}_{1} \in {A}_{0} \) . As in the proof of Theorem 2, define a colouring \( {c}_{1} : {B}_{1}^{\left( r... | Yes |
Theorem 4. Let \( T \) be a tree of order \( t \) . Then \( r\left( {{K}^{s}, T}\right) = \left( {s - 1}\right) \left( {t - 1}\right) + 1 \) . | Proof. The graph \( \left( {s - 1}\right) {K}^{t - 1} \) does not contain \( T \), its complement, \( {K}_{s - 1}\left( {t - 1}\right) \), does not contain \( {K}^{s} \), so \( r\left( {{K}^{s}, T}\right) \geq \left( {s - 1}\right) \left( {t - 1}\right) + 1 \) . Let now \( G \) be a graph of order \( \left( {s - 1}\rig... | Yes |
Lemma 5. \( r\left( {G,{H}_{1} \cup {H}_{2}}\right) \leq \max \left\{ {r\left( {G,{H}_{1}}\right) + \left| {H}_{2}\right|, r\left( {G,{H}_{2}}\right) }\right\} \) . In particular, \( r\left( {s{H}_{1},{H}_{2}}\right) \leq r\left( {{H}_{1},{H}_{2}}\right) + \left( {s - 1}\right) \left| {H}_{1}\right| . \) | Proof. Let \( n \) be greater than the right hand side and suppose there is a red-blue colouring of \( {K}^{n} \) without a red \( G \) . Then \( n \geq r\left( {G,{H}_{2}}\right) \) implies that there is a blue \( {H}_{2} \) . Remove it. Since \( n - \left| {H}_{2}\right| \geq r\left( {G,{H}_{1}}\right) \), the remain... | Yes |
Theorem 6. If \( s \geq t \geq 1 \) then\n\n\[ r\left( {s{K}^{2}, t{K}^{2}}\right) = {2s} + t - 1. \] | Proof. The graph \( G = {K}^{{2s} - 1} \cup {E}^{t - 1} \) does not contain \( s \) independent edges and \( \bar{G} = {E}^{{2s} - 1} + {K}^{t - 1} \) does not contain \( t \) independent edges. Hence \( r\left( {s{K}^{2}, t{K}^{2}}\right) \geq {2s} + t - 1. \)\n\nTrivially \( r\left( {s{K}^{2},{K}^{2}}\right) = {2s} \... | Yes |
Theorem 7. If \( s \geq t \geq 1 \) and \( s \geq 2 \) then \( r\left( {s{K}^{3}, t{K}^{3}}\right) = {3s} + {2t} \) . | Proof. Let \( G = {K}^{{3s} - 1} \cup \left( {{K}^{1} + {E}^{{2t} - 1}}\right) \) . Then \( G \) does not contain \( s \) independent triangles and \( \bar{G} = {E}^{{3s} - 1} + \left( {{K}^{1} \cup {K}^{{2t} - 1}}\right) \) does not contain \( t \) independent triangles. Hence \( r\left( {s{K}^{3}, t{K}^{3}}\right) \)... | Yes |
Theorem 8. If \( s \geq t \geq 1 \) then\n\n\[ \n{ps} + \left( {q - 1}\right) t - 1 \leq r\left( {s{K}^{p}, t{K}^{q}}\right) \leq {ps} + \left( {q - 1}\right) t + C.\n\] | Proof. The graph \( {K}^{{ps} - 1} \cup {E}^{\left( {q - 1}\right) t - 1} \) shows the first inequality. As in the proofs of the previous theorems, we fix \( s - t \) and apply induction on \( t \) . By Lemma 5 we have\n\n\[ \nr\left( {s{K}^{p}, t{K}^{q}}\right) \leq \left( {s - t}\right) p + r\left( {t{K}^{p}, t{K}^{q... | Yes |
Theorem 10. \( M \rightarrow {\left( \mathcal{P}\right) }_{k} \) iff there is a finite set \( X \subset M \) such that \( X \rightarrow {\left( \mathcal{P}\right) }_{k} \) . | Proof. If \( M \nrightarrow {\left( \mathcal{P}\right) }_{k} \) then clearly \( X \nrightarrow {\left( \mathcal{P}\right) }_{k} \) for every finite set \( X \) . Given a finite set \( X \subset M \), let \( N\left( X\right) \) be the set of colourings in which \( X \) does not contain a monochromatic set \( P \in \math... | Yes |
Theorem 11. Let \( P \) be a pair of points at distance 1 apart. Then \[ {\mathbb{R}}^{2} \rightarrow {\left( P\right) }_{3}\;\text{ but }\;{\mathbb{R}}^{2} \nrightarrow {\left( P\right) }_{7}. \] | Proof. Figure VI. 2 shows the first assertion. For suppose that in a red-blue-yellow colouring of the seven points there is no monochromatic adjacent pair. We may assume that \( x \) is red. Then \( {y}_{1},{z}_{1} \) are blue and yellow so \( {x}_{1} \) is red. Similarly \( {x}_{2} \) is red, but \( {x}_{1} \) and \( ... | No |
Theorem 12. Let \( {Q}^{2} \) be the (vertex set of a) unit square. Then \( {\mathbb{R}}^{6} \rightarrow {\left( {Q}^{2}\right) }_{2} \) . | Proof. Consider a red-blue colouring of \( {\mathbb{R}}^{6} \) . Then, in particular, we have a red-blue colouring of the following fifteen points of \( {\mathbb{R}}^{6} : {x}_{ij} = \left( {{x}_{ij}^{1},\ldots ,{x}_{ij}^{6}}\right) \) , \( 1 \leq i < j \leq 6,{x}_{ij}^{k} = 0 \) unless \( k = i \) or \( j \) and \( {x... | Yes |
Theorem 13. If \( {L}_{1} \) and \( {L}_{2} \) are Ramsey then so is \( {L}_{1} \times {L}_{2} \) . | Proof. Given \( k \), there is an \( m \) such that \( {\mathbb{R}}^{m} \rightarrow {\left( {L}_{1}\right) }_{k} \) . Hence by the compactness theorem there is a finite subset \( X \subset {\mathbb{R}}^{m} \) such that \( X \rightarrow {\left( {L}_{1}\right) }_{k} \) . Put \( l = {k}^{\left| X\right| } \) . Since \( {L... | Yes |
Theorem 15. Let \( S \) be a commutative infinite semigroup. Let \( L \) be a finite subset of \( S \) and let \( \mathcal{P} = \{ a + {nL} : a \in S, n \in \mathbb{N}\} \), where \( {nL} = \{ {nx} : x \in L\} \). Then\n\n\[ S \rightarrow {\left( \mathcal{P}\right) }_{k}\;\text{ for every }k \] | The proof is too difficult to present in these notes. However, we state this theorem partly because it is the cornerstone of very general results proved by Graham, Leeb and Rothschild, asserting that certain categories are \ | No |
Lemma 18. If \( \mathbb{N} \) rejects \( \varnothing \) then there exists an \( M \in {N}^{\left( \omega \right) } \) that rejects every \( X \subset M \) . | Proof. Note first that there is an \( {M}_{0} \) such that every \( X \subset {M}_{0} \) is either accepted or rejected by \( {M}_{0} \) . Indeed, put \( {N}_{0} = \mathbb{N},{a}_{0} = 1 \) . Suppose that we have defined \( {N}_{0} \supset {N}_{1} \supset \cdots \supset {N}_{k} \) and \( {a}_{i} \in {N}_{i} - {N}_{i + ... | Yes |
Theorem 19. Every open subset of \( {2}^{\mathbb{N}} \) is Ramsey. | Proof. Let \( \mathcal{F} \subset {2}^{\mathbb{N}} \) be open and assume that \( {A}^{\left( \omega \right) } ⊄ \mathcal{F} \) for every \( A \in {N}^{\left( \omega \right) } \) , i.e., \( \mathbb{N} \) rejects \( \varnothing \) . Let \( M \) be the set whose existence is guaranteed by Lemma 18. If \( {M}^{\left( \omeg... | Yes |
Corollary 20. Let \( {\mathcal{F}}_{0} \subset {N}^{\left( < \omega \right) } \) be dense. Then there is an \( M \in {N}^{\left( \omega \right) } \) such that every \( A \subset M \) has an initial segment belonging to \( {\mathcal{F}}_{0} \) . | Proof. Let \( \mathcal{F} = \left\{ {F \subset \mathbb{N} : F}\right. \) has an initial segment belonging to \( \left. {\mathcal{F}}_{0}\right\} \) . Then \( \mathcal{F} \) is open so there is an \( M \in {\mathbb{N}}^{\left( \omega \right) } \) such that either \( {M}^{\left( \omega \right) } \subset \mathcal{F} \), i... | Yes |
Corollary 21. Let \( {\mathcal{F}}_{0} \subset {N}^{\left( < \omega \right) } \) be a thin family. Then for any \( k \) -colouring of \( {\mathcal{F}}_{0} \) there is an infinite set \( A \subset \mathbb{N} \) such that all members of \( {\mathcal{F}}_{0} \) contained in \( A \) have the same colour. | Proof. It suffices to prove the result for \( k = 2 \) . Consider a red and blue colouring of \( {\mathcal{F}}_{0} : {\mathcal{F}}_{0} = {\mathcal{F}}_{\text{red }} \cup {\mathcal{F}}_{\text{blue }} \) . If \( {\mathcal{F}}_{\text{red }} \) is dense then let \( M \) be the set guaranteed by Corollary 20. For every \( F... | Yes |
Theorem 1. The expected number of \( {K}^{s} \) subgraphs contained in a graph \( G \in \Omega = \mathcal{G}\left( {n, m}\right) \) is\n\n\[ E\left( {X}_{s}\right) = \left( \begin{array}{l} n \\ s \end{array}\right) \left( \begin{array}{l} N - \left( \begin{array}{l} s \\ 2 \end{array}\right) \\ M - \left( \begin{array... | Proof. As we noted earlier, \( \left| \Omega \right| = \left( \begin{matrix} N \\ M \end{matrix}\right) \) . In order to calculate the expected number of \( {K}^{s} \) subgraphs in \( G \in \Omega \), first we compute the number of graphs \( G \in \Omega \) that contain a fixed complete subgraph \( {K}_{0} \) of order ... | Yes |
Theorem 2. If \( s, t \geq 3 \) then\n\n\[ R\left( {s, t}\right) > \exp \left\{ \frac{\left( {s - 1}\right) \left( {t - 1}\right) }{2\left( {s + t}\right) }\right\} \]\n\nIn particular\n\n\[ R\left( {s, s}\right) > {e}^{\left( {1/4}\right) {\left( s - 1}\right) }^{2}/s}. \] | Proof. Note first that the graph \( G = \left( {t - 1}\right) {K}^{s - 1} \) does not contain a \( {K}^{s} \) and its complement, \( {K}_{s - 1}\left( {t - 1}\right) \), does not contain a \( {K}^{t} \), so \( R\left( {s, t}\right) \geq \) \( \left( {s - 1}\right) \left( {t - 1}\right) + 1 \) . Therefore we may assume ... | Yes |
Theorem 3. Let \( 2 \leq s \leq {n}_{1},2 \leq t \leq {n}_{2},\alpha = \left( {s - 1}\right) /\left( {{st} - 1}\right) \) and \( \beta = \) \( \left( {t - 1}\right) /\left( {{st} - 1}\right) \) . Then there is a bipartite graph \( {G}_{2}\left( {{n}_{1},{n}_{2}}\right) \) of size\n\n\[ \left\lfloor {\left( {1 - \frac{1... | Proof. Let\n\n\[ n = {n}_{1} + {n}_{2} \]\n\n\[ {V}_{1} = \left\{ {1,2,\ldots ,{n}_{1}}\right\} \]\n\n\[ {V}_{2} = \left\{ {{n}_{1} + 1,{n}_{1} + 2,\ldots ,{n}_{1} + {n}_{2}}\right\} \]\n\n\[ E = \left\{ {{ij} : i \in {V}_{1}, j \in {V}_{2}}\right\} \]\n\n\[ M = \left\lfloor {{n}_{1}^{1 - \alpha }{n}_{2}^{1 - \beta }}\... | Yes |
Theorem 4. Given natural numbers \( \delta \geq 3 \) and \( g \geq 4 \) there is a graph of order at most \( {\left( 2\delta \right) }^{g} \) whose minimum degree is at least \( \delta \) and whose girth is at least \( g \) . | Proof. Let \( \Omega = \mathcal{G}\left( {n, M}\right) \), where \( n = {\left( 2\delta \right) }^{g} \) and \( M = {\delta n} \) . Denote by \( {Z}_{l}\left( G\right) \) the number of cycles of length \( l \) in a graph \( G \) ; thus \( {Z}_{l} \) is a random variable on the space \( \Omega \) . What is \( E\left( {Z... | Yes |
Theorem 6. There is a constant \( c > 0 \) such that\n\n\[ R\left( {3, t}\right) \geq c{\left( \frac{t}{\log t}\right) }^{2} \]\n\nfor every \( t \geq 2 \) . | It suffices to show that there is a (large) positive constant \( A \) such that for every \( n \) there is a graph \( {G}^{n} \) without triangles and without \( \left\lfloor {A{n}^{1/2}\log n}\right\rfloor \) independent vertices. Indeed, it is easily checked that if there is such a constant then \( c = \frac{1}{4}{A}... | Yes |
Theorem 7. Let \( 1 \leq h \leq k \) be fixed natural numbers and let \( 0 < p < 1 \) be fixed also. Then in \( \mathcal{G}\left( {n, P\left( \text{edge}\right) = p}\right) \) a.e. graph \( G \) is such that for every sequence of \( k \) vertices \( {x}_{1},{x}_{2},\ldots ,{x}_{k} \) there exists a vertex \( x \) such ... | Proof. Let \( {x}_{1},{x}_{2},\ldots ,{x}_{k} \) be an arbitrary sequence. The probability that a vertex \( x \in W = V\left( G\right) - \left\{ {{x}_{1},\ldots ,{x}_{k}}\right\} \) has the required properties is \( {p}^{h}{q}^{k - h} \) . Since for \( x, y \in W, x \neq y \), the edges \( x{x}_{i} \) are chosen indepe... | Yes |
Theorem 9. If \( 0 < p < 1 \) is fixed then in \( \Omega = \mathcal{G}\left( {n, P\left( \text{edge}\right) = p}\right) \) the maximal degree of almost every graph is\n\n\[{pn} + {\left( 2pqn\log n\right) }^{1/2} + o{\left( n\log n\right) }^{1/2}.\] | Proof. Let \( c > 0 \) and denote by \( {X}_{c} = {X}_{c}\left( G\right) \) the number of vertices of degree at least \( d\left( c\right) = \lfloor {pn} + c{\left( pqn\log n\right) }^{1/2}\rfloor \) . Then, by (10),\n\n\[{\mu }_{c} = E\left( {X}_{c}\right) = n\mathop{\sum }\limits_{{k = d\left( c\right) }}^{{n - 1}}\le... | Yes |
Corollary 10. If \( 0 < p < 1 \) and \( c > \sqrt{}\left( {3/2}\right) \) then in \( \mathcal{G}\left( {n, P\left( \text{edge}\right) = p}\right) \) almost no graph has two vertices of equal degree whose degrees are at least \[ d\left( c\right) \doteq \left\lfloor {{pn} + c{\left( pqn\log n\right) }^{1/2}}\right\rfloor... | Proof. The probability of the existence of two such vertices is at most \[ {n}^{2}\mathop{\sum }\limits_{{k = d\left( c\right) }}^{{n - 1}}{\left\{ \left( \begin{array}{l} n - 2 \\ k - 1 \end{array}\right) {p}^{k - 1}{q}^{n - 1 - k}\right\} }^{2} \] and by (11) this is \( o\left( 1\right) \) . | Yes |
Theorem 11. Let \( k \geq 2, k - 1 \leq l \leq \left( \begin{array}{l} k \\ 2 \end{array}\right) \) and let \( F = G\left( {k, l}\right) \) be a graph with average degree \( \left( {{2l}/k}\right) \) at least as large as that of any of its subgraphs. Then \( {n}^{-k/l} \) is a threshold function for \( F \), that is if... | Proof. Let \( p = \gamma {n}^{-k/l},0 < \gamma < {n}^{k/l} \), and denote by \( X = X\left( G\right) \) the number of copies of \( F \) contained in \( G \in \mathcal{G}\left( {n, P\left( \text{edge}\right) = p}\right) \) . Denote by \( {k}_{F} \) the number of graphs with a fixed set of \( k \) labelled vertices that ... | Yes |
Theorem 12. Let \( 0 < p < 1 \) be fixed. Then the clique number of almost every graph \( G \in \mathcal{G}\left( {n, P\left( \text{edge}\right) = p}\right) \) is \( \lfloor d\rfloor \) or \( \lceil d\rceil \) . | Proof. The assertion is equivalent to the following:\n\n\[ P\left( {{X}_{r} > 0}\right) \rightarrow 0\;\text{ if }r \geq d + 1 \]\n\n\[ P\left( {{X}_{r} > 0}\right) \rightarrow 1\;\text{ if }r \leq d - 1 \]\n\nNow if \( r \geq d + 1 \), then by (6),\n\n\[ E\left( {X}_{r}\right) = \left( \begin{array}{l} n \\ r \end{arr... | Yes |
Lemma 13. Let \( c > 3 \) and \( 0 < \gamma < \frac{1}{3} \) be constants and let \( p = \left( {c\log n}\right) /n \) . Then in \( \mathcal{G}\left( {n, P\left( \text{edge}\right) = p}\right) \) we have\n\n\[ P\left( {{D}_{t} > 0}\right. \text{for some}\left. {t,1 \leq t \leq {\gamma n}}\right) = O\left( {n}^{3 - c}\r... | Proof. Put \( \beta = \left( {c - 3}\right) /{4c} \) . Clearly\n\n\[ \mathop{\sum }\limits_{{t = 1}}^{{\lfloor {\gamma n}\rfloor }}E\left( {D}_{t}\right) = \mathop{\sum }\limits_{{t = 1}}^{{\lfloor {\gamma n}\rfloor }}\left( \begin{array}{l} n \\ t \end{array}\right) \left( \begin{matrix} n - t \\ n - {3t} \end{matrix}... | Yes |
Theorem 2. The subgroup \( B \) of \( A \) is generated by the decorations of the chords. | In particular, the subgroup \( B \) in Figure VIII. 9 is generated by \( b, a{b}^{3}{a}^{-1} \) , \( a{b}^{-1}a{b}^{-1}{a}^{-1} \) and \( {abab}{a}^{-1} \). | Yes |
Theorem 3. The subgroup \( B \) has a presentation\n\n\[ \left\langle {{a}_{1},{b}_{1},\ldots ,{a}_{2},{b}_{2},\ldots \mid {R}_{\mu }^{1},{R}_{v}^{1},\ldots ,{R}_{\mu }^{2},{R}_{v}^{2},\ldots }\right\rangle \] | Now if we wish to preserve the connection between this presentation of \( B \) and the original presentation of \( A \), we simply equate \( {c}_{i} \) with the decoration of the edge coloured \( c \) starting at vertex \( i \) .\n\nIf \( A \) is a free group, its presentation contains no relations. Hence the above pre... | Yes |
Theorem 4. A subgroup of a free group is free. Furthermore, if \( A \) is a free group of rank \( k \) (that is it has \( k \) free generators) and \( B \) is a subgroup of index \( n \) then \( B \) has rank \( \left( {k - 1}\right) n + 1 \) . | Proof. The presentation of \( B \) given in Theorem 3 is a free presentation on the set of chords of the Schreier diagram. Altogether there are \( {kn} \) edges of which \( n - 1 \) are tree edges; hence there are \( \left( {k - 1}\right) n + 1 \) chords. | Yes |
Theorem 5. Let \( G \) be a connected graph of order \( n \) with adjacency matrix \( A \) . (i) Every eigenvalue \( \lambda \) of \( G \) satisfies \( \left| \lambda \right| \leq \Delta = \Delta \left( G\right) \) . | Proof. (i) Let \( \mathbf{x} = \left( {x}_{i}\right) \) be a non-zero eigenvector with eigenvalue \( \lambda \) . Let \( {x}_{p} \) be a weight with maximum modulus: \( \left| {x}_{p}\right| \geq \left| {x}_{i}\right| \) for every \( i \) ; we may assume without loss of generality that \( {x}_{p} = 1 \) . Then \[ \left... | Yes |
Theorem 7. Let \( G \) be a connected non-bipartite highly regular graph of order \( n \) with collapsed adjacency matrix \( C \) . Let \( {\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{r} \) be the roots of the characteristic polynomial of \( C \) different from \( k \), the degree of the vertices of G. Then there ... | Proof. We know from Theorem 5 that \( {\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{r} \) are the eigenvalues of \( A \) in addition to \( k \), which has multiplicity 1 . Thus if \( m\left( {\lambda }_{i}\right) \) is the multiplicity of \( {\lambda }_{i} \) then \[ 1 + \mathop{\sum }\limits_{{i = 1}}^{r}m\left( {... | Yes |
Theorem 8. If there is a strongly regular graph of order \( n \) with parameters \( \left( {k, a, b}\right) \) then\n\n\[ \n{m}_{1},{m}_{2} = \frac{1}{2}\left\{ {n - 1 \pm \frac{\left( {n - 1}\right) \left( {b - a}\right) - {2k}}{{\left\{ {\left( a - b\right) }^{2} + 4\left( k - b\right) \right\} }^{1/2}}}\right\} \n\]... | Proof. The characteristic polynomial of the collapsed adjacency matrix \( \mathrm{C} \) is\n\n\[ \n{x}^{3} + \left( {b - a - k}\right) {x}^{2} + \left( {\left( {a - b}\right) k + b - k}\right) x + k\left( {k - b}\right) .\n\]\n\nOn dividing by \( x - k \), we find that the roots different from \( k \) are\n\n\[ \n{\lam... | Yes |
Theorem 9. Suppose there is a \( k \) -regular graph \( G \) of order \( n = {k}^{2} + 1 \) and diameter 2. Then \( k = 2,3,7 \) or 57 . | Proof. We know from Theorem 1 of Chapter IV that \( G \) is strongly regular with parameters \( \left( {k,0,1}\right) \) . By the rationality condition at least one of the following two conditions has to hold:\n\n(i) \( \left( {n - 1}\right) - {2k} = {k}^{2} - {2k} = 0 \) and \( n - 1 = {k}^{2} \) is even,\n\n(ii) \( 1... | Yes |
Theorem 10. There are \( {n}^{n - 2} \) trees on \( n \) labelled vertices. | Proof. As in Chapter VII, let \( V = \{ 1,2,\ldots, n\} \) be the set of vertices. Given a tree \( T \), associate a code with \( T \) as follows. Remove the endvertex with the smallest label and write down the label of the adjacent vertex. Repeat the process until only two vertices remain. The code obtained is a seque... | No |
Corollary 11. Let \( {d}_{1} \leq {d}_{2} \leq \cdots \leq {d}_{n} \) be the degree sequence of a tree: \( {d}_{1} \geq 1 \) and \( \mathop{\sum }\limits_{{i = 1}}^{n}{d}_{i} = {2n} - 2 \) . Then the number of labelled trees of order \( n \) with degree sequence \( {\left( {d}_{i}\right) }_{1}^{n} \) is given by the mu... | \[ \left( \begin{matrix} n - 2 \\ {d}_{1} - 1,{d}_{2} - 1,\ldots ,{d}_{n} - 1 \end{matrix}\right) \] | Yes |
Lemma 12. \( \left| \Gamma \right| \mathop{\sum }\limits_{{i = 1}}^{l}w\left( {O}_{i}\right) = \mathop{\sum }\limits_{{alpha \in \Gamma }}\mathop{\sum }\limits_{{x \in F\left( \alpha \right) }}w\left( x\right) \) . | Proof.\n\n\[ \mathop{\sum }\limits_{{alpha \in \Gamma }}\mathop{\sum }\limits_{{x \in F\left( \alpha \right) }}w\left( x\right) = \mathop{\sum }\limits_{{x \in X}}\mathop{\sum }\limits_{{alpha \in \Gamma \left( x\right) }}w\left( x\right) = \mathop{\sum }\limits_{{i = 1}}^{l}\mathop{\sum }\limits_{{x \in {O}_{i}}}\math... | Yes |
Let \( X = \{ 1,2,3,4\} \) and \( \Gamma = \{ 1,\left( {12}\right) ,\left( {34}\right) ,\left( {12}\right) \left( {34}\right) \} \) . What is \( N\left( \Gamma \right) \)? | Clearly \( F\left( 1\right) = \{ 1,2,3,4\}, F\left( \left( {12}\right) \right) = \{ 3,4\}, F\left( \left( {34}\right) \right) = \{ 1,2\} \) and \( \left. {F(\left( {12}\right) \left( {34}\right) }\right) = \varnothing \) . Thus \( N\left( \Gamma \right) = \frac{1}{4}\{ 4 + 2 + 2 + 0\} = 2. \) | No |
Consider all bracelets made up of 5 beads. The beads can be red, blue and green, and two bracelets are considered to be identical if one can be obtained from the other by rotation. (Reflections are not allowed!) How many distinct bracelets are there? | In this case we choose \( X \) to be the set of all \( {3}^{5} = {243} \) bracelets and let \( \Gamma \) be \( {C}_{5} \), the cyclic group of order 5, acting on \( X \) . Then the question is: how many orbits does \( \Gamma \) have? For the identity \( 1 \in \Gamma \) clearly \( F\left( 1\right) = X \) . For every non... | Yes |
Theorem 13. (Pólya's enumeration theorem)\n\n\[ \left| \Gamma \right| S = \widetilde{Z}\left( {\Gamma ;{s}_{1},{s}_{2},\ldots ,{s}_{d}}\right) \] | Proof. By Lemma 12\n\n\[ \left| \Gamma \right| S = \left| \Gamma \right| \mathop{\sum }\limits_{{i = 1}}^{l}w\left( {O}_{i}\right) = \mathop{\sum }\limits_{{\alpha \in \Gamma }}\mathop{\sum }\limits_{{f \in F\left( {\alpha }^{ * }\right) }}w\left( f\right) . \]\n\nNow clearly\n\n\( F\left( {\alpha }^{ * }\right) = \lef... | Yes |
What happens if in the previous example we allow reflections? Then \( \Gamma \) is the dihedral group \( {D}_{5} \), the group of symmetries of the regular pentagon, whose cycle sum is \( {a}_{1}^{5} + 4{a}_{5} + 5{a}_{1}{a}_{2}^{2} \) . | Thus if we take, as before, \( A = \mathbb{Z}\left\lbrack {x, y}\right\rbrack, w\left( r\right) = 1, w\left( b\right) = x \) and \( w\left( g\right) = y \), we find that the number of bracelets containing 2 red,1 blue and 2 green beads is the coefficient of \( x{y}^{2} \) in \( \frac{1}{10}\left\{ {{\left( 1 + x + y\ri... | Yes |
Place 3 red, 2 blue and 1 yellow balls in the 6 vertices of an octahedron. In how many distinct ways can this be done? | The group of symmetries of the octahedron has cycle sum \( {a}_{1}^{6} + 6{a}_{1}^{2}{a}_{4} + 3{a}_{1}^{2}{a}_{2}^{2} + 6{a}_{2}^{3} + 8{a}_{3}^{2} : {a}_{1}^{6} \) comes from the identity, \( 6{a}_{1}^{2}{a}_{4} \) from rotations through \( \pi \) about axes through the vertices, \( 6{a}_{2}^{3} \) from rotations thr... | Yes |
Example 6. Place red, blue, green and yellow balls into the vertices of an octahedron. Denote by \( {P}_{i} \) the set of patterns in which the total number of red and blue balls is congruent to \( i \) modulo 4 . What is \( \left| {P}_{0}\right| - \left| {P}_{2}\right| \) ? | The cycle sum of the rotation group of the octahedron was calculated in Exercise 5 and was found to be \( {a}_{1}^{6} + 6{a}_{1}^{2}{a}_{4} + 3{a}_{1}^{2}{a}_{2}^{2} + 6{a}_{2}^{3} + 8{a}_{3}^{2} \) . Let \( A = \mathbb{C} \), the field of complex numbers, and put \( w\left( r\right) = w\left( b\right) = i \) , \( w\le... | Yes |
Let \( {\mathfrak{M}}_{k, n}\left( \mathbf{R}\right) \) be the \( k \times n \) matrices with real coefficients. Let \( {M}_{k, n}\left( \mathbf{R}\right) \) be the \( k \times n \) matrices of rank \( k\left( {k \leq n}\right) \). Let \( {M}_{k, n}^{m}\left( \mathbf{R}\right) \) be the elements of \( {\mathfrak{M}}_{k... | For convenience we delete the \( \mathbf{R} \) and refer to \( {M}_{k, n}^{m} \). For \( {X}_{0} \in {M}_{k, n}^{m} \), we define a coordinate neighborhood at \( {X}_{0} \) as follows. Since the rank of \( X \) is \( m \), there exist permutation matrices \( P, Q \) such that \( P{X}_{0}Q = \left\lbrack \begin{array}{l... | Yes |
Example 1.8 (Grassmannian manifolds): Let \( V \) be a finite dimensional \( K \) -vector space and let \( {G}_{k}\left( V\right) \mathrel{\text{:=}} \{ \) the set of \( k \) -dimensional subspaces of \( V\} \) , for \( k < {\dim }_{K}V \) . Such a \( {G}_{k}\left( V\right) \) is called a Grassmannian manifold. We shal... | Consider, for example, \( {G}_{k, n}\left( \mathbf{R}\right) \) . We can define the map \[ \pi : {M}_{k, n}\left( \mathbf{R}\right) \rightarrow {G}_{k, n}\left( \mathbf{R}\right) \] where \[ \pi \left( A\right) = \pi \left( \begin{matrix} {a}_{1} \\ \cdot \\ \cdot \\ \cdot \\ {a}_{k} \end{matrix}\right) \mathrel{\text{... | Yes |
Consider \( {\mathbf{P}}_{n} = {\mathbf{P}}_{n}\left( \mathbf{C}\right) \), and let\n\n\[ H = \left\{ {\left\lbrack {{z}_{0},\ldots ,{z}_{n}}\right\rbrack \in {\mathbf{P}}_{n} : {a}_{0}{z}_{0} + \cdots + {a}_{n}{z}_{n} = 0}\right\} ,\]\n\nwhere \( \left( {{a}_{0},\ldots ,{a}_{n}}\right) \in {\mathbf{C}}^{n + 1} - \{ 0\... | Let \( {U}_{\alpha } \) be the coordinate systems for \( {\mathbf{P}}_{n} \) as defined in Example 1.6. Let us consider \( {U}_{0} \cap H \) , and let \( \left( {{\zeta }_{1},\ldots ,{\zeta }_{n}}\right) \) be coordinates in \( {\mathbf{C}}^{n} \) . Suppose that \( \left\lbrack {{z}_{0},\ldots ,{z}_{n}}\right\rbrack \i... | Yes |
Let \( M \) be a differentiable manifold. Then we want to construct a vector bundle over \( M \) whose fibre at each point is the linearization of the manifold \( M \), to be called the tangent bundle to \( M \) . Let \( p \in M \) . Then we let\n\n\[ \n{\mathcal{E}}_{M, p} \mathrel{\text{:=}} \mathop{\lim }\limits_{\s... | Since \( M \) is a differentiable manifold, we can find a diffeomorphism \( h \) defined in a neighborhood \( U \) of \( p \) where\n\n\[ \nh : U \rightarrow {U}^{\prime }\underset{\text{open }}{ \subset }{\mathbf{R}}^{n} \n\]\n\nand where, letting \( {h}^{ * }f\left( x\right) = f \circ h\left( x\right), h \) has the p... | Yes |
Example 2.6 (Universal bundle): Let \( {U}_{r, n} \) be the disjoint union of the \( r \) -planes ( \( r \) -dimensional \( K \) -linear subspaces) in \( {K}^{n} \) . Then there is a natural projection\n\n\[ \pi : {U}_{r, n} \rightarrow {G}_{r, n} \]\n\nwhere \( {G}_{r, n} = {G}_{r, n}\left( K\right) \), given by \( \p... | Letting \( {U}_{\alpha } = \left\{ {\left\lbrack {{x}_{0},\ldots ,{x}_{n - 1}}\right\rbrack \in {\mathbf{P}}_{n - 1} : {x}_{\alpha } \neq 0}\right\} \) ,(cf. Example 1.6), we see that\n\n\[ {\pi }^{-1}\left( {U}_{\alpha }\right) = \left\{ {v = t\left( {{x}_{0},\ldots ,{x}_{n - 1}}\right) \in {\mathbf{R}}^{n} : t \in \m... | Yes |
Theorem 3.15 (de Rham): Let \( X \) be a differentiable manifold. Then the natural mapping \[ I : {H}^{p}\left( {{\mathcal{E}}^{ * }\left( X\right) }\right) \rightarrow {H}^{p}\left( {{\mathcal{S}}_{\infty }^{ * }\left( {X,\mathbf{R}}\right) }\right) \] induced by integration of differential forms over \( {\mathbf{C}}^... | Proof: As in Example 2.14, consider the resolutions of \( \mathbf{R} \) given by  Then the sheaves \( {\mathcal{E}}^{ * } \) and \( {\mathcal{S}}_{\infty }^{ * } \) are both soft. Since \( {\mathcal{E}}^{ * } \) is fin... | No |
Theorem 3.17 (Dolbeault): Let \( X \) be a complex manifold. Then\n\n\[ \n{H}^{q}\left( {X,{\mathbf{\Omega }}^{p}}\right) \cong \frac{\operatorname{Ker}\left( {{\mathcal{E}}^{p, q}\left( X\right) \overset{\bar{\partial }}{ \rightarrow }{\mathcal{E}}^{p, q + 1}\left( X\right) }\right) }{\operatorname{Im}\left( {{\mathca... | Proof: The resolution given in Example 2.12 is a fine resolution, and we can apply Theorem 3.13.\n\nQ.E.D. | No |
Proposition 1.3 (Sobolev): Let \( f \) be a measurable \( {L}^{2} \) function in \( {\mathbf{R}}^{n} \) with \( \parallel f{\parallel }_{s} < \infty \), for \( s > \left\lbrack {n/2}\right\rbrack + k + 1 \), a nonnegative integer. Then \( f \in {C}^{k}\left( {\mathbf{R}}^{n}\right) \) (after a possible change on a set ... | Proof: Our assumption \( \parallel f{\parallel }_{s} < \infty \) means that\n\n\[ \n{\int }_{{\mathbf{R}}^{n}}{\left| \widehat{f}\left( \xi \right) \right| }^{2}{\left( 1 + {\left| \xi \right| }^{2}\right) }^{s}{d\xi } < \infty .\n\]\n\nLet\n\n\[ \n\widetilde{f}\left( x\right) = {\int }_{{\mathbf{R}}^{n}}{e}^{i\langle ... | Yes |
Theorem 2.5 (Poincaré duality): Let \( X \) be a compact \( m \) -dimensional ori-entable differentiable manifold. Then there is a conjugate linear isomorphism\n\n\[ \n\sigma : {H}^{r}\left( {X,\mathbf{C}}\right) \rightarrow {H}^{m - r}\left( {X,\mathbf{C}}\right) ,\n\]\n\nand hence \( {H}^{m - r}\left( {X,\mathbf{C}}\... | Proof: Introduce a Riemannian metric and an orientation on \( X \) and let * be the associated \( * \) -operator. Then we have the commutative diagram \n\nwhere \( {H}_{\Delta } \) is the projection onto the harmonic... | Yes |
Theorem 2.7 (Serre duality): Let \( X \) be a compact complex manifold of complex dimension \( n \) and let \( E \rightarrow X \) be a holomorphic vector bundle over \( X \) . Then there is a conjugate linear isomorphism\n\n\[ \n\sigma : {H}^{r}\left( {X,{\mathbf{\Omega }}^{p}\left( E\right) }\right) \rightarrow {H}^{n... | Proof: By introducing Hermitian metrics on \( X \) and \( E \), we can define the \( {\bar{ * }}_{E} \) operator. Then we obtain the following commutative diagram, \n\nwhich proves the result immediately. Once again,... | Yes |
Theorem 6.7 (Griffiths): The period mapping (6.7) is a holomorphic mapping. | Remark: The proof of this theorem depends principally on the Kodaira-Spencer deformation theory formalism (Kodaira and Spencer [1]), which we do not develop here (see e.g., Morrow and Kodaira [1]). | No |
For any given field \( \mathbf{k} \), there is a category denoted Vect \( {}_{\mathbf{k}} \) whose objects \( V, W,\ldots \) are vector spaces over \( \mathbf{k} \) and whose morphisms are linear transformations. | To verify the claim, suppose \( T : V \rightarrow W \) and \( S : W \rightarrow U \) are linear transformations. Then for any \( v,{v}^{\prime } \in V \) and any \( k \in \mathbf{k} \) ,\n\n\[ \n{ST}\left( {{kv} + {v}^{\prime }}\right) = S\left( {{kTv} + T{v}^{\prime }}\right) = {kSTv} + {ST}{v}^{\prime } \n\]\n\nand s... | No |
A metric space is called complete if every Cauchy sequence converges. Being a complete metric space is not a topological property. | For instance, the map \( \left( {-1,1}\right) \rightarrow \) \( \mathbb{R} \) by \( x \mapsto \frac{x}{\left( 1 - {x}^{2}\right) } \) is a homeomorphism, yet \( \mathbb{R} \) is a complete metric space while \( \left( {-1,1}\right) \) is not. This example also shows that being bounded is also not a topological property... | Yes |
Theorem 0.1 The following are equivalent.\n\n- \( f : X \rightarrow Y \) is an isomorphism.\n\n- For every object \( Z \), the pushforward \( {f}_{ * } : \mathrm{C}\left( {Z, X}\right) \rightarrow \mathrm{C}\left( {Z, Y}\right) \) is an isomorphism of sets.\n\n- For every object \( Z \), the pullback \( {f}^{ * } : \ma... | Proof. We’ll prove that a morphism \( f : X \rightarrow Y \) is an isomorphism if and only if for every object \( Z \), the pushforward \( {f}_{ * } : \mathrm{C}\left( {Z, X}\right) \rightarrow \mathrm{C}\left( {Z, Y}\right) \) is an isomorphism of sets; then, we’ll leave the other statements as exercises.\n\nSuppose \... | No |
Two sets \( X \) and \( Y \) are isomorphic if and only if \( \operatorname{Set}\left( {Z, X}\right) \cong \operatorname{Set}\left( {Z, Y}\right) \) for all sets \( Z \). | That is, \( X \) and \( Y \) are the same if and only if they relate to all other sets in the same way. But this is overkill! Two sets are isomorphic if and only if they have the same cardinality, so to distinguish \( X \) and \( Y \) we need only look at the case when \( Z \) is the one-point set \( * \) . Indeed, a m... | Yes |
Unless otherwise specified, the natural numbers \( \mathbb{N} \) and the integers \( \mathbb{Z} \) are given discrete topologies, but there are others. Notably, there is a topology on \( \mathbb{Z} \) for which the sets\n\n\[ S\left( {a, b}\right) = \{ {an} + b \mid n \in \mathbb{N}\} \]\n\nfor \( a \in \mathbb{Z} \sma... | It's not hard to check that the sets \( S\left( {a, b}\right) \) are also closed in this topology, and since every integer except \( \pm 1 \) has a prime factor, it follows that\n\n\[ \mathbb{Z} \smallsetminus \{ - 1, + 1\} = \mathop{\bigcup }\limits_{{p\text{ prime }}}S\left( {p,0}\right) \]\n\nSince the left hand sid... | Yes |
Example 1.8 We can generalize the previous example from \( {\mathbb{R}}^{n} \) to \( {\mathbb{R}}^{\mathbb{N}} \), the space of sequences in \( \mathbb{R} \), if we avoid those sequences with divergent norm. The set \( {l}_{p} \) of sequences \( \left\{ {x}_{n}\right\} \) for which \( \mathop{\sum }\limits_{{n = 1}}^{\... | \[{\begin{Vmatrix}\left\{ {x}_{i}\right\} \end{Vmatrix}}_{p}\; \mathrel{\text{:=}} \;{\left( \mathop{\sum }\limits_{{i = 1}}^{\infty }{\left| {x}_{i}\right| }^{p}\right) }^{\frac{1}{p}}\] | Yes |
Theorem 1.2 Let \( X \) be a topological space, let \( S \) be a set, and let \( \pi : X \rightarrow S \) be surjective. The quotient topology on \( S \) is determined by the following property.\n\nUniversal property for the quotient topology For every topological space \( Z \) and every function \( f : S \rightarrow Z... | Proof. Exercise. | No |
The map \( \pi : \left\lbrack {0,1}\right\rbrack \rightarrow {S}^{1} \) defined by \( \pi \left( t\right) = \left( {\cos \left( {2\pi t}\right) ,\sin \left( {2\pi t}\right) }\right) \) is a quotient map. | Therefore, for any space \( Z \), continuous functions \( {S}^{1} \rightarrow Z \) are the same as continuous functions \( \left\lbrack {0,1}\right\rbrack \rightarrow Z \) which factor through \( \pi \) . That is, continuous functions \( {S}^{1} \rightarrow Z \) are the same as paths \( \gamma : \left\lbrack {0,1}\righ... | No |
Theorem 1.3 Let \( {\left\{ {X}_{\alpha }\right\} }_{\alpha \in A} \) be an arbitrary collection of topological spaces, and let \( X = \) \( \mathop{\prod }\limits_{{\alpha \in A}}{X}_{\alpha } \) . Let \( {\pi }_{\alpha } : X \rightarrow {X}_{\alpha } \) denote the natural projection. The product topology on \( X \) i... | Proof. Exercise. | No |
Example 1.19 Let \( X = {\mathbb{R}}^{2} \) . One can write any function \( f : S \rightarrow {\mathbb{R}}^{2} \) in terms of component functions \( {fs} = \left( {{xs},{ys}}\right) \), where the components \( {xs} \) and \( {ys} \) are simply given by the composition\n\n![bac6c572-2793-477b-b8ba-f26676893837_32_1.jpg]... | But be careful: functions from \( {\mathbb{R}}^{2} \) and more generally \( {\mathbb{R}}^{n} \) can be confusing, in part because our familiarity with \( {\mathbb{R}}^{n} \) can give unjustified topological importance to the maps \( \mathbb{R} \rightarrow {\mathbb{R}}^{2} \) given by fixing one of the coordinates. So d... | Yes |
Theorem 1.4 Let \( {\left\{ {X}_{\alpha }\right\} }_{\alpha \in A} \) be an arbitrary collection of topological spaces and let \( X = \) \( \mathop{\coprod }\limits_{{\alpha \in A}} \) . Let \( {i}_{\alpha } : {X}_{\alpha } \rightarrow X \) denote the natural inclusion. The coproduct topology on \( X \) is characterize... | Proof. Exercise. | No |
Example 1.20 Any set \( X \) is the coproduct over its points viewed as singletons:\n\n\[ X \cong \mathop{\coprod }\limits_{{x \in X}}\{ x\} \] | As topological spaces, however, \( X \) is homeomorphic to \( \mathop{\coprod }\limits_{{x \in X}}\{ x\} \) if and only if \( X \) has the discrete topology. | No |
The space \( {\mathbb{R}}^{n} \) is homotopic to the one-point space, \( {\mathbb{R}}^{n} \simeq * \) . | To see this, define \( f : * \rightarrow {\mathbb{R}}^{n} \) by \( * \mapsto 0 \) and define \( g : {\mathbb{R}}^{n} \rightarrow * \) in the only way possible. Then \( {gf} = {\operatorname{id}}_{ * } \), and \( {fg} : {\mathbb{R}}^{n} \rightarrow 0 \) is homotopic to \( {\operatorname{id}}_{{\mathbb{R}}^{n}} \) via th... | Yes |
Theorem 2.1 If \( X \) is (path) connected and \( f : X \rightarrow Y \), then \( {fX} \) is (path) connected. | Proof. If \( {fX} \) is not connected, then there is a nonconstant map \( g : {fX} \rightarrow \{ 0,1\} \), which implies the map \( {gf} : X \rightarrow \{ 0,1\} \) is not constant. Now suppose \( X \) is path connected. Let \( y,{y}^{\prime } \in {fX} \) so that \( y = {fx} \) and \( {y}^{\prime } = f{x}^{\prime } \)... | Yes |
Corollary 2.1.1 Connected and path connected are topological properties. | Since quotient maps are continuous surjections, we know quotients preserve (path) connectedness. | No |
Theorem 2.2 Let \( X \) be a space and \( f : X \rightarrow Y \) be a surjective map. If \( Y \) is connected in the quotient topology and if each fiber \( {f}^{-1}y \) is connected, then \( X \) is connected. | Proof. Let \( g : X \rightarrow \{ 0,1\} \) . Since the fibers of \( f \) are connected, \( g \) must be constant on each fiber of \( f \) . Therefore \( g \) factors through \( f : X \rightarrow Y \), and there is a map \( \bar{g} : Y \rightarrow \{ 0,1\} \) that fits into this diagram.\n\n![bac6c572-2793-477b-b8ba-f2... | Yes |
Theorem 2.3 Suppose \( X = \mathop{\bigcup }\limits_{{\alpha \in A}}{X}_{\alpha } \) and that for each \( \alpha \in A \) the space \( {X}_{\alpha } \) is (path) connected. If there is a point \( x \in \mathop{\bigcap }\limits_{{\alpha \in A}}{X}_{\alpha } \) then \( X \) is (path) connected. | Proof. Exercise. | No |
Example 2.1 The rational numbers \( \mathbb{Q} \) are not connected as the continuous map \( k : \mathbb{Q} \rightarrow \) \( \{ 0,1\} \) defined by \( {kx} = 0 \) if \( x < \sqrt{2} \) and \( {kx} = 1 \) if \( x > \sqrt{2} \) shows. | In fact, the rationals are totally disconnected, meaning that the only connected subsets are singletons. | Yes |
Theorem 2.4 The connected subspaces of \( \mathbb{R} \) are intervals. | Proof. Suppose \( A \) is a connected subspace of \( \mathbb{R} \) that is not an interval. Then there exist \( x, y \in A \) such that \( x < z < y \) for some \( z \notin A \) . Thus\n\n\[ A = \left( {A \cap \left( {-\infty, z}\right) }\right) \cup \left( {A \cap \left( {z,\infty }\right) }\right) \]\n\nis a separati... | Yes |
Theorem 2.5 Path connected implies connected. | Proof. Suppose \( X \) is path connected, and let \( k : X \rightarrow \{ 0,1\} \) be a function. Choose any two points in \( X \) . There exists a path connecting them. Since \( k \) must be constant on that path, it takes the same value at these two points. Therefore \( k \) is constant. | Yes |
Theorem 2.6 Connected and path connected are homotopy invariants. | Proof. Suppose \( f : X \rightarrow Y \) is a homotopy equivalence, and let \( g : Y \rightarrow X \) and \( h : Y \times I \rightarrow Y \) be a homotopy from \( {fg} \) to \( {\mathrm{{id}}}_{Y} \) .\n\nSuppose that \( X \) is connected. To show that \( Y \) is connected, let \( k : Y \rightarrow \{ 0,1\} \) be any m... | Yes |
Theorem 2.7 Every convex polygon can be partitioned into two convex polygons, each having the same area and same perimeter. | Proof. Let \( P \) be a convex polygon, and first observe that finding a line that bisects the area of \( P \) is not difficult. Simply take a vertical line and consider the difference of the area on the left and the right. As the line moves from left to right the difference goes from negative to positive continuously ... | Yes |
Theorem 2.8 Every continuous function \( f : \left\lbrack {-1,1}\right\rbrack \rightarrow \left\lbrack {-1,1}\right\rbrack \) has a fixed point. | Proof. Suppose \( f : \left\lbrack {-1,1}\right\rbrack \rightarrow \left\lbrack {-1,1}\right\rbrack \) is a continuous function for which \( {fx} \neq x \) for all \( x \in \left\lbrack {-1,1}\right\rbrack \) . In particular we have \( f\left( {-1}\right) > - 1 \) and \( {f1} < 1 \) . Now define a map \( g : \left\lbra... | Yes |
Theorem 2.9 Let \( {\left\{ {X}_{\alpha }\right\} }_{\alpha \in A} \) be a collection of (path) connected topological spaces. Then \( X \mathrel{\text{:=}} \mathop{\prod }\limits_{{\alpha \in A}}{X}_{\alpha } \) is (path) connected. | Proof. We'll prove the theorem for path connected spaces and will leave the rest as an exercise. Suppose \( {X}_{\alpha } \) is path connected for every \( \alpha \in A \), and let \( a, b \in X \) . Since each \( {X}_{\alpha } \) is path connected, there exists a path \( {p}_{\alpha } : \left\lbrack {0,1}\right\rbrack... | No |
Theorem 2.12 In any locally path connected topological space, the connected components and path components are the same. | Proof. Exercise. | No |
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