A hypergraph blowup lemma

Transcription

1 A hypergraph blowup lemma Peter Keevash School of Mathematical Sciences, Queen Mary, University of London.

2 Introduction Extremal problems Given F, what conditions on G guarantee F G?

3 Introduction Extremal problems Given F, what conditions on G guarantee F G? Fixed configurations Mantel: Largest triangle-free graph is complete bipartite.

4 Introduction Extremal problems Given F, what conditions on G guarantee F G? Fixed configurations Mantel: Largest triangle-free graph is complete bipartite. Turán: Largest tetrahedron-free 3-graph?

5 Introduction Extremal problems Given F, what conditions on G guarantee F G? Fixed configurations Mantel: Largest triangle-free graph is complete bipartite. Turán: Largest tetrahedron-free 3-graph? Spanning configurations Dirac: Graph on [n] with min degree n/2 is Hamiltonian.

6 Introduction Extremal problems Given F, what conditions on G guarantee F G? Fixed configurations Mantel: Largest triangle-free graph is complete bipartite. Turán: Largest tetrahedron-free 3-graph? Spanning configurations Dirac: Graph on [n] with min degree n/2 is Hamiltonian. Rödl-Ruciński-Szemerédi: k-graph on [n] with min (k 1)-degree (1/2 + o(1))n has a tight Hamilton cycle.

7 Szemerédi s Regularity Lemma

8 Regularity properties

9 The graph blowup lemma

10 The graph blowup lemma (Komlós, Sárközy and Szemerédi) Super-regular pairs are equivalent to corresponding complete bipartite graphs for embedding bounded degree subgraphs.

11 Regular and quasirandom 3-complexes

12 Regular approximation

13 Regular approximation Regular approximation theorem (Rödl and Schacht) Any k-graph H can be loosely approximated by some G highly regular wrt some moderately complex P.

14 The hypergraph blowup lemma

15 The hypergraph blowup lemma Hypergraph blowup lemma (K.) Super-regular k-complexes are equivalent to corresponding complete k-complexes for embedding bounded degree subhypergraphs.

16 Applications Theorem (K.) Suppose H is k-graph on [n], all degrees H(x) = (1 ± ɛ)c 1 ( n 1 k 1), all codegrees H(S) > c 2 n, and F is fixed unbalanced k-partite k-graph, with 1/n ɛ 1/C c 1, c 2, 1/k. Then H contains F-packing covering all but < C vertices.

17 Applications Theorem (K.) Suppose H is k-graph on [n], all degrees H(x) = (1 ± ɛ)c 1 ( n 1 k 1), all codegrees H(S) > c 2 n, and F is fixed unbalanced k-partite k-graph, with 1/n ɛ 1/C c 1, c 2, 1/k. Then H contains F-packing covering all but < C vertices. Theorem (K., Kühn, Mycroft, Osthus / Hàn, Schacht) Any k-graph on [n] with min (k 1)-degree ( 1 2(k 1) + o(1) ) n has a loose Hamilton cycle.

18 Applications Theorem (K.) Suppose H is k-graph on [n], all degrees H(x) = (1 ± ɛ)c 1 ( n 1 k 1), all codegrees H(S) > c 2 n, and F is fixed unbalanced k-partite k-graph, with 1/n ɛ 1/C c 1, c 2, 1/k. Then H contains F-packing covering all but < C vertices. Theorem (K., Kühn, Mycroft, Osthus / Hàn, Schacht) Any k-graph on [n] with min (k 1)-degree ( 1 2(k 1) + o(1) ) n has a loose Hamilton cycle. Example S [n], S = n 2(k 1) 1, H = {E ( [n]) k : E S }.

19 The random greedy algorithm Problem Setup (special case: r-partite, no restricted positions): r-partite k-graph H max degree on X = r i=1 X i, (G, M) super-regular on V = r i=1 V i, X i = V i = G i.

20 The random greedy algorithm Problem Setup (special case: r-partite, no restricted positions): r-partite k-graph H max degree on X = r i=1 X i, (G, M) super-regular on V = r i=1 V i, X i = V i = G i. Find φ : X V with φ(h) G \ M.

21 The random greedy algorithm Problem Setup (special case: r-partite, no restricted positions): r-partite k-graph H max degree on X = r i=1 X i, (G, M) super-regular on V = r i=1 V i, X i = V i = G i. Find φ : X V with φ(h) G \ M. Algorithm Initialisation: Buffer B V (H) mutual distance 9, N neighbourhoods of b B, R = V (H) \ (B N), list L = NRB, queue q =.

22 The random greedy algorithm Problem Setup (special case: r-partite, no restricted positions): r-partite k-graph H max degree on X = r i=1 X i, (G, M) super-regular on V = r i=1 V i, X i = V i = G i. Find φ : X V with φ(h) G \ M. Algorithm Initialisation: Buffer B V (H) mutual distance 9, N neighbourhoods of b B, R = V (H) \ (B N), list L = NRB, queue q =. Iteration: (i) Select x to embed (jump > queue > list) (ii) Embed x φ(x) random good vertex (iii) Add problem vertices to queue (some jump) (iv) Stop when all non-buffer vertices embedded.

23 The random greedy algorithm Problem Setup (special case: r-partite, no restricted positions): r-partite k-graph H max degree on X = r i=1 X i, (G, M) super-regular on V = r i=1 V i, X i = V i = G i. Find φ : X V with φ(h) G \ M. Algorithm Initialisation: Buffer B V (H) mutual distance 9, N neighbourhoods of b B, R = V (H) \ (B N), list L = NRB, queue q =. Iteration: (i) Select x to embed (jump > queue > list) (ii) Embed x φ(x) random good vertex (iii) Add problem vertices to queue (some jump) (iv) Stop when all non-buffer vertices embedded. Conclusion: Complete embedding with system of distinct representatives on available slots for unembedded vertices.

24 The random greedy algorithm Problem Setup (special case: r-partite, no restricted positions): r-partite k-graph H max degree on X = r i=1 X i, (G, M) super-regular on V = r i=1 V i, X i = V i = G i. Find φ : X V with φ(h) G \ M. Algorithm Initialisation: Buffer B V (H) mutual distance 9, N neighbourhoods of b B, R = V (H) \ (B N), list L = NRB, queue q =. Iteration: (i) Select x to embed (jump > queue > list) (ii) Embed x φ(x) random good vertex (iii) Add problem vertices to queue (some jump) (iv) Stop when all non-buffer vertices embedded. Conclusion: Complete embedding with system of distinct representatives on available slots for unembedded vertices. Theorem The random greedy algorithm embeds H in G \ M with probability 1 o(1).

25 Embeddings of complexes Complex-coloured complexes F(t) = {F S (t)} S H : (i) available: F S (t) G S, all P F S (t) contain embedded part φ(s e t ) (ii) compatible: F S (t) = S SF S (t) is a complex

26 Embeddings of complexes Complex-coloured complexes F(t) = {F S (t)} S H : (i) available: F S (t) G S, all P F S (t) contain embedded part φ(s e t ) (ii) compatible: F S (t) = S SF S (t) is a complex Update rule (for φ(x) = y) F S (t) = F S {x} (t 1)(y) ( if S {x} H) = {P V S : P {y} F S {x} (t 1)}.

27 Embeddings of complexes Complex-coloured complexes F(t) = {F S (t)} S H : (i) available: F S (t) G S, all P F S (t) contain embedded part φ(s e t ) (ii) compatible: F S (t) = S SF S (t) is a complex Update rule (for φ(x) = y) F S (t) = F S {x} (t 1)(y) ( if S {x} H) = {P V S : P {y} F S {x} (t 1)}. C(S) = F S {x}(t 1)(y), S S,S {x} H F S (t) = F S (t 1)[C(S)] \ y.

28 Super-regularity

29 Conclusion Any k-graph can be roughly approximated by very regular pieces. Low degree combinations of these pieces = super-regular (universal for low degree embedding) + small trash.

30 Conclusion Any k-graph can be roughly approximated by very regular pieces. Low degree combinations of these pieces = super-regular (universal for low degree embedding) + small trash. Super-regularity is a complicated definition, but natural simplifications fail on certain examples.

31 Conclusion Any k-graph can be roughly approximated by very regular pieces. Low degree combinations of these pieces = super-regular (universal for low degree embedding) + small trash. Super-regularity is a complicated definition, but natural simplifications fail on certain examples. Many potential applications to hypergraph generalisations of results that used the graph blowup lemma.