An asymptotic multipartite Kühn-Osthus theorem

Transcription

1 An asymptotic multipartite Kühn-Osthus theorem Ryan R. Martin 1 Richard Mycroft 2 Jozef Skokan 3 1 Iowa State University 2 University of Birmingham 3 London School of Economics 08 August 2017 Algebraic and Extremal Graph Theory University of Delaware, Newark, DE Martin s research partially supported by: NSF grant DMS , NSA grant H , Simons Foundation grant # and an Iowa State University Faculty Development Grant. Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

2 Joint Work This talk is based on joint work with: Richard Mycroft University of Birmingham Jozef Skokan London School of Economics Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

3 The Hajnal-Szemerédi theorem Theorem (Hajnal-Szemerédi, 1970) (Complementary form) If G is a simple graph on n vertices with minimum degree ( δ(g) 1 1 ) n k then G contains a subgraph which consists of n/k vertex-disjoint copies of K k. artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

4 The Hajnal-Szemerédi theorem Theorem (Hajnal-Szemerédi, 1970) (Complementary form) If G is a simple graph on n vertices with minimum degree ( δ(g) 1 1 ) n k then G contains a subgraph which consists of n/k vertex-disjoint copies of K k. This is a K k -tiling. Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

5 The Hajnal-Szemerédi theorem Theorem (Hajnal-Szemerédi, 1970) (Complementary form) If G is a simple graph on n vertices with minimum degree ( δ(g) 1 1 ) n k then G contains a subgraph which consists of n/k vertex-disjoint copies of K k. This is a K k -tiling or a K k -factor or even a K k -packing. Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

6 The Hajnal-Szemerédi theorem Theorem (Hajnal-Szemerédi, 1970) (Complementary form) If G is a simple graph on n vertices with minimum degree ( δ(g) 1 1 ) n k then G contains a subgraph which consists of n/k vertex-disjoint copies of K k. This is a K k -tiling or a K k -factor or even a K k -packing. We will use tiling most often. Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

7 The Hajnal-Szemerédi theorem Theorem (Hajnal-Szemerédi, 1970) (Complementary form) If G is a simple graph on n vertices with minimum degree ( δ(g) 1 1 ) n k then G contains a subgraph which consists of n/k vertex-disjoint copies of K k. This is a K k -tiling or a K k -factor or even a K k -packing. We will use tiling most often. Notes k = 2 follows from Dirac Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

8 The Hajnal-Szemerédi theorem Theorem (Hajnal-Szemerédi, 1970) (Complementary form) If G is a simple graph on n vertices with minimum degree ( δ(g) 1 1 ) n k then G contains a subgraph which consists of n/k vertex-disjoint copies of K k. This is a K k -tiling or a K k -factor or even a K k -packing. We will use tiling most often. Notes k = 2 follows from Dirac k = 3 proven by Corrádi & Hajnal 1963 Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

9 The Alon-Yuster theorem Theorem (Alon-Yuster, 1992) For any α > 0 and graph H, there exists an n 0 = n 0 (α, H) such that in any graph G on n n 0 vertices with ( δ(g) 1 1 ) n + αn χ(h) there is an H-tiling of G if V (H) divides n. Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

10 The Alon-Yuster theorem Theorem (Alon-Yuster, 1992) For any α > 0 and graph H, there exists an n 0 = n 0 (α, H) such that in any graph G on n n 0 vertices with ( δ(g) 1 1 ) n + αn χ(h) there is an H-tiling of G if V (H) divides n. Komlós, Sárközy and Szemerédi, 2001, showed that αn can be replaced by C = C(H), but not eliminated entirely. Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

11 The Alon-Yuster theorem Theorem (Alon-Yuster, 1992) For any α > 0 and graph H, there exists an n 0 = n 0 (α, H) such that in any graph G on n n 0 vertices with ( δ(g) 1 1 ) n + αn χ(h) there is an H-tiling of G if V (H) divides n. Theorem (Kühn-Osthus, 2009) For any graph H, there exists an n 0 = n 0 (H) and a constant C = C(H) such that in any graph G on n n 0 vertices with ( δ(g) 1 1 ) χ n + C (H) there is an H-tiling of G if V (H) divides n. Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

12 The Alon-Yuster theorem Theorem (Kühn-Osthus, 2009) For any graph H, there exists an n 0 = n 0 (H) and a constant C = C(H) such that in any graph G on n n 0 vertices with ( δ(g) 1 1 ) χ n + C (H) there is an H-tiling of G if V (H) divides n. This result is best possible, up to the constant C. But what is χ (H)? Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

13 Critical chromatic number Definition Let H be a graph with order: h = V (H) chromatic number: χ = χ(h) artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

14 Critical chromatic number Definition Let H be a graph with order: h = V (H) chromatic number: χ = χ(h) σ = σ(h) is the order of the smallest color class of H among all proper χ-colorings of V (H). artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

15 Critical chromatic number Definition Let H be a graph with order: h = V (H) chromatic number: χ = χ(h) σ = σ(h) is the order of the smallest color class of H among all proper χ-colorings of V (H). The critical chromatic number of H, is χ cr = χ cr (H) = (χ 1)h h σ artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

16 Critical chromatic number Definition Let H be a graph with order: h = V (H) chromatic number: χ = χ(h) σ = σ(h) is the order of the smallest color class of H among all proper χ-colorings of V (H). The critical chromatic number of H, is χ cr = χ cr (H) = (χ 1)h h σ Fact For any graph H: χ(h) 1 < χ cr (H) χ(h) artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

17 Critical chromatic number Definition Let H be a graph with order: h = V (H) chromatic number: χ = χ(h) σ = σ(h) is the order of the smallest color class of H among all proper χ-colorings of V (H). The critical chromatic number of H, is χ cr = χ cr (H) = (χ 1)h h σ Fact For any graph H: χ(h) 1 < χ cr (H) χ(h) Also, χ cr (H) = χ(h) iff every proper χ-coloring of H is a equipartition. χ cr (H) was defined by Komlós, artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

18 Critical chromatic number Definition Let H be a graph with order: h = V (H) chromatic number: χ = χ(h) σ = σ(h) is the order of the smallest color class of H among all proper χ-colorings of V (H). The critical chromatic number of H, is χ cr = χ cr (H) = (χ 1)h h σ { χ χcr (H), if gcd(h) = 1; (H) = χ(h), else. where gcd(h) is basically the gcd of the differences of the color classes in proper colorings of H. artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

19 Definitions Definition The family of k-partite graphs with n vertices in each part is denoted G k (n). Definition The natural bipartite subgraphs of G are the ones induced by the pairs of classes of the k-partition. Definition If G G k (n), let ˆδ k (G) denote the minimum degree among all of the natural bipartite subgraphs of G. Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

20 Multipartite Hajnal-Szemerédi The asymptotic Hajnal-Szemerédi theorem was solved with two different methods: Theorem (Keevash-Mycroft, 2013; Lo-Markström, 2013) Let k 2 and ɛ > 0. There exists an n 0 = n 0 (k, ɛ) such that if n n 0, G G k (n) and if ( ˆδ k (G) 1 1 ) n + ɛn, k then G has a K k -tiling. Hypergraph blow-up; Absorbing method Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

21 Multipartite Hajnal-Szemerédi The asymptotic Hajnal-Szemerédi theorem was solved with two different methods: Theorem (Keevash-Mycroft, 2013; Lo-Markström, 2013) Let k 2 and ɛ > 0. There exists an n 0 = n 0 (k, ɛ) such that if n n 0, G G k (n) and if ( ˆδ k (G) 1 1 ) n + ɛn, k then G has a K k -tiling. Hypergraph blow-up; Absorbing method Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

22 Multipartite Hajnal-Szemerédi The asymptotic Hajnal-Szemerédi theorem was solved with two different methods: Theorem (Keevash-Mycroft, 2013; Lo-Markström, 2013) Let k 2 and ɛ > 0. There exists an n 0 = n 0 (k, ɛ) such that if n n 0, G G k (n) and if ( ˆδ k (G) 1 1 ) n + ɛn, k then G has a K k -tiling. Hypergraph blow-up; Absorbing method Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

23 Multipartite Hajnal-Szemerédi In a longer manuscript, Keevash and Mycroft settle the multipartite Hajnal-Szemerédi case for large n: Theorem (Keevash-Mycroft, 2013, Mem. Amer. Math. Soc.) Let k 2 and ɛ > 0. There exists an n 0 = n 0 (k, ɛ) such that if n n 0, G G k (n) and if ( ˆδ k (G) 1 1 ) n, k then G has a K k -tiling or both k and n/k are odd integers and G Γ k (n/k). The case of k = 3 was solved by Magyar-M. (2002). The case of k = 4 was solved by M.-Szemerédi (2008). The graph Γ k (n/k) is one of Catlin s Type 2 graphs. Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

24 Catlin s Type 2 Graphs Catlin s Type 2 graph. The red indicates non-edges between graph classes. Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

25 Toward Kühn-Osthus Theorem (Zhao, 2009) Let h be a positive integer. There exists an n 0 = n 0 (h) such that if n n 0, h n, and G G 2 (n) with δ(g) = ˆδ 2 (G) { 1 2 then G has a perfect K h,h -tiling. n + h 1, if n/h is odd; 1 2 n + 3h 2 2, if n/h is even, Moreover, there are examples that prove that this ˆδ 2 condition cannot be improved. Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

26 Toward Kühn-Osthus Theorem (Bush-Zhao, 2012) Let H be a bipartite graph. There exists an n 0 = n 0 (H) and c = c(h) such that if n n 0, V (H) n, and G G 2 (n) with ( ) 1 1 χ δ(g) (H) n +c, if gcd(h) = 1 or gcd cc (H) > 1; ( ) 1 1 χ(h) n +c, if gcd(h) > 1 and gcd cc (H) = 1, then G has a perfect H-tiling. The quantity gcd cc (H) counts the gcd of the sizes of the connected components of H. Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

27 Our results Theorem (M.-Skokan, 2013+) Let k 2, H be a graph with χ(h) = k and ɛ > 0. There exists an n 0 = n 0 (H, ɛ) such that if n n 0, G G k (n) and if ( ˆδ k (G) 1 1 ) n + ɛn, χ(h) then G has an H-tiling. Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

28 Our results Theorem (M.-Skokan, 2013+) Let k 2, H be a graph with χ(h) = k and ɛ > 0. There exists an n 0 = n 0 (H, ɛ) such that if n n 0, G G k (n) and if ( ˆδ k (G) 1 1 ) n + ɛn, χ(h) then G has an H-tiling. This, of course, contains the asymptotic Hajnal-Szemerédi case. Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

29 Our results Theorem (M.-Mycroft-Skokan, 2015+) Let k 2, H be a graph with χ(h) = k, χ = χ (H) and ɛ > 0. There exists an n 0 = n 0 (H, ɛ) such that if n n 0, G G k (n) and if ( ˆδ k (G) 1 1 ) χ n + ɛn, (H) then G has an H-tiling. The main tool is linear programming. Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

30 Linear programming Definition For any graph G, let T k (G) denote the set of k-cliques of G. The fractional K k -tiling number, τk (G) is: max w(t ) τk (G) = s.t. T T k (G) T T k (G),T v w(t ) 0, w(t ) 1, v V (G), T T k (G). artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

31 Linear programming Definition For any graph G, let T k (G) denote the set of k-cliques of G. The fractional K k -tiling number, τk (G) is: max w(t ) τk (G) = s.t. T T k (G) T T k (G),T v w(t ) 0, w(t ) 1, v V (G), T T k (G). Theorem Let k 2. If G G k (n) and ˆδ k (G) (k 1)n/k, then τk (G) = n. artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

32 Linear programming Definition τk (G) = max s.t. w(t ) T T k (G) w(t ) 1, T T k (G),T v v V (G), w(t ) 0, T T k (G). Theorem Let k 2. If G G k (n) and ˆδ k (G) (k 1)n/k, then τk (G) = n. The proof is by induction on k and uses both the Duality Theorem and Complementary Slackness Theorem of LPs. artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

33 Linear programming Theorem Let k 2. If G G k (n) and ˆδ k (G) (k 1)n/k, then τk (G) = n. The proof is by induction on k and uses both the Duality Theorem and Complementary Slackness Theorem of LPs. Duality Theorem: max w(t ) τk (G) = s.t. w(t ) 1, v, T v w(t ) 0, T. min = s.t. x(v) x(v) 1, T, v T x(v) 0, v. artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

34 Linear programming Theorem Let k 2. If G G k (n) and ˆδ k (G) (k 1)n/k, then τk (G) = n. The proof is by induction on k and uses both the Duality Theorem and Complementary Slackness Theorem of LPs. Duality Theorem: max w(t ) τk (G) = s.t. w(t ) 1, v, T v w(t ) 0, T. min = s.t. x(v) x(v) 1, T, v T x(v) 0, v. UB: τk (G) n. Setting x(v) 1/k gives a feasible solution to the minlp, so τk (G) (kn) (1/k) = n. artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

35 Linear programming Theorem Let k 2. If G G k (n) and ˆδ k (G) (k 1)n/k, then τk (G) = n. max w(t ) τk (G) = s.t. w(t ) 1, v, T v w(t ) 0, T. min = s.t. x(v) x(v) 1, T, v T x(v) 0, v. UB: τk (G) n. Setting x(v) 1/k gives a feasible solution to the minlp, so τk (G) (kn) (1/k) = n. LB: τk (G) n. Base Case: k = 2. artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

36 Linear programming max w(t ) τk (G) = s.t. w(t ) 1, v, T v w(t ) 0, T. min = s.t. x(v) x(v) 1, T, v T x(v) 0, v. UB: τk (G) n. Setting x(v) 1/k gives a feasible solution to the minlp, so τk (G) (kn) (1/k) = n. LB: τk (G) n. Base Case: k = 2. Let G = (V 1, V 2 ; E). If either V 1 or V 2 fails to have a slack vertex in the maxlp, then τk (G) w(t ) = w(t ) = 1 = n. T v V i T v v V i artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

37 Linear programming max w(t ) τk (G) = s.t. w(t ) 1, v, T v w(t ) 0, T. min = s.t. x(v) x(v) 1, T, v T x(v) 0, v. LB: τk (G) n. Base Case: k = 2. Let G = (V 1, V 2 ; E). If either V 1 or V 2 fails to have a slack vertex in the maxlp, then τk (G) w(t ) = w(t ) = 1 = n. T v V i T v v V i If v 1 V 1 and v 2 V 2 are slack, then we may assume x(v 1 ) = x(v 2 ) = 0 (Complementary Slackness). artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

38 Linear programming max w(t ) τk (G) = s.t. w(t ) 1, v, T v w(t ) 0, T. min = s.t. x(v) x(v) 1, T, v T x(v) 0, v. LB: τk (G) n. Base Case: k = 2. Let G = (V 1, V 2 ; E). If either V 1 or V 2 fails to have a slack vertex in the maxlp, then τk (G) w(t ) = w(t ) = 1 = n. T v V i T v v V i If v 1 V 1 and v 2 V 2 are slack, then we may assume x(v 1 ) = x(v 2 ) = 0 (Complementary Slackness). Each vertex in N(v 1 ), N(v 2 ) has weight 1. Since N(v 1 ), N(v 2 ) n/2, τk (G) n. artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

39 Linear programming max w(t ) τk (G) = s.t. w(t ) 1, v, T v w(t ) 0, T. min = s.t. x(v) x(v) 1, T, v T x(v) 0, v. LB: τk (G) n. Induction Step Let G = (V 1,..., V k ; E). If any V i has no slack vertices in the maxlp, then τk (G) w(t ) = w(t ) = 1 = n. T v V i T v v V i artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

40 Linear programming max w(t ) τk (G) = s.t. w(t ) 1, v, T v w(t ) 0, T. min = s.t. x(v) x(v) 1, T, v T x(v) 0, v. LB: τk (G) n. Induction Step Let G = (V 1,..., V k ; E). If any V i has no slack vertices in the maxlp, then τk (G) w(t ) = w(t ) = 1 = n. T v V i T v v V i If v i V i, i, are slack, then we may assume x(v i ) = 0, i. artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

41 Linear programming max w(t ) τk (G) = s.t. w(t ) 1, v, T v w(t ) 0, T. min = s.t. x(v) x(v) 1, T, v T x(v) 0, v. LB: τk (G) n. Induction Step Let G = (V 1,..., V k ; E). If any V i has no slack vertices in the maxlp, then τk (G) w(t ) = w(t ) = 1 = n. T v V i T v v V i If v i V i, i, are slack, then we may assume x(v i ) = 0, i. Let G i G[N(v i )], i, so that G i has exactly k 1 k n vertices in each V j. artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

42 Linear programming LB: τk (G) n. Induction Step Let G = (V 1,..., V k ; E). If any V i has no slack vertices in the maxlp, then τk (G) w(t ) = w(t ) = 1 = n. T v V i T v v V i If v i V i, i, are slack, then we may assume x(v i ) = 0, i. Let G i G[N(v i )], i, so that G i has exactly k 1 k n vertices in each V j. Each G i satisfies the degree requirement for G k 1 ( k 1 k n). artin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

43 Linear programming LB: τk (G) n. Induction Step Let G = (V 1,..., V k ; E). If any V i has no slack vertices in the maxlp, then τk (G) w(t ) = w(t ) = 1 = n. T v V i T v v V i If v i V i, i, are slack, then we may assume x(v i ) = 0, i. Let G i G[N(v i )], i, so that G i has exactly k 1 k n vertices in each V j. Each G i satisfies the degree requirement for G k 1 ( k 1 k n). By induction, k (k 1)τk (G) x(v) i=1 v V (G i ) k i=1 k 1 n = (k 1)n. k Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

44 Linear programming LB: τk (G) n. Induction Step Let G = (V 1,..., V k ; E). If any V i has no slack vertices in the maxlp, then τk (G) w(t ) = w(t ) = 1 = n. T v V i T v v V i If v i V i, i, are slack, then we may assume x(v i ) = 0, i. Let G i G[N(v i )], i, so that G i has exactly k 1 k n vertices in each V j. Each G i satisfies the degree requirement for G k 1 ( k 1 k n). By induction, k (k 1)τk (G) x(v) i=1 v V (G i ) k i=1 k 1 n = (k 1)n. k Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

45 Future work Can we replace ˆδ k (G) ( ˆδ k (G) 1 1 χ (G) ( ) 1 1 χ (G) n + ɛn with ) n + C(H)? Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

46 Future work Can we replace ˆδ k (G) ( ˆδ k (G) 1 1 χ (G) ( ) 1 1 χ (G) n + ɛn with ) n + C(H)? Is ˆδ k (G) (k 1)n/k + ɛn sufficient to force the k th power of a Hamilton cycle? (Related to Bollobás-Komlós conjecture on bandwidth) Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

47 Future work Can we replace ˆδ k (G) ( ˆδ k (G) 1 1 χ (G) ( ) 1 1 χ (G) n + ɛn with ) n + C(H)? Is ˆδ k (G) (k 1)n/k + ɛn sufficient to force the k th power of a Hamilton cycle? (Related to Bollobás-Komlós conjecture on bandwidth) What probability p guarantees that, for any G with ˆδ k (G) (k 1)n/k + ɛn, the random subgraph G p has a K k -tiling? Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13

48 Thanks! My home page: My CV (with links to this and previous talks): Contact me: Martin (Iowa State University University ofan Birmingham asymptotic multipartite London School Kühn-Osthus of Economics) theorem 08 August / 13