Turán numbers of expanded hypergraph forests

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1 Turán numbers of expanded forests Rényi Institute of Mathematics, Budapest, Hungary Probabilistic and Extremal Combinatorics, IMA, Minneapolis, Sept. 9, Main results are joint with Tao JIANG, Miami University, Oxford, OH, USA

2 Turán type problems Given a k-uniform hypergraph A. What is the max number of edges of F, F ( [n]) k, if A F? Notation of this threshold: ex (k) (n,a) := max F. E.g., the classical Turán theorem on max triangle free graphs: ex(n,k 3 ) = 1 4 n2. The complete bipartite graph K n/2, n/2 is extremal. In general: (Turán 1941) e(g n ) > e(t n,p ) = K p+1 G n and here T n,p := the Turán graph, is the unique extremal graph.

3 Basics about graphs: The three zones theorem Theorem (Erdős, Stone, Simonovits, Kővári, T. Sós, Turán) If G is any graph and n then one of the three cases holds G is a forest and ex(n,g) = O(n), G has cycles, but it is bipartite, then for some c := c(g) > 0 one has Ω(n 1+c ) < ex(n,g) < O(n 2 c ), χ(g) = p + 1 3, then ( ex(n,g) = (1+o(1)) 1 1 )( ) n. p 2 QUESTION: What is the situation for k-uniform hypergraphs? A modest partial answer: We have asymptotic for some tree-like hypergraphs.

4 Trees and forests in graphs Fact ( If T is a tree (forest) of v vertices then ) ex(n,t) (v 2)n. Proof. Leave out degrees (v 2). If an H is left with δ(h) k 1, then T can be embedded into H. Lower bound: disjoint complete graphs on v 1 vertices. No P 6 : ex(n,t v ) v 2 n O(v 2 ). 2 For every forest F with e(f) > 1 we have ex(n,f) = O(n).

5 Trees in graphs; the Erdős-Sós conjecture Theorem (Ajtai, Komlós, Simonovits, Szemerédi 2014+) If T is a tree of v vertices, v v 0, then ex(n,t) 1 (v 2)n. 2 Infinitely difficult (over 180 pages). Other versions by Ajtai, Hladky, Komlós, Piquet, Simonovits, Szemerédi Why is it so difficult? Probably because there are many different (almost) extremal constructions.

6 Paths in graphs Theorem (Erdős-Gallai, If P v is a path of v vertices then) ex(n,p v ) 1 (v 2)n No P 6. Another extremal graph for even v! Exact ex(n,p v ) by Faudree & Schelp 1976/ Kopylov 1977.

7 Forests in hypergraphs What is the situation for k-uniform hypergraphs? A recursive definition: A single k-set is a k-forest. Suppose that T = {E 1,E 2,...,E u } ( V k) is a k-forest and A := A u+1 E i for some 1 i u, and B V =, A + B = k. Then {E 1,E 2,...,E u,e u+1 } is a k-forest with E u+1 := A B. k = 2 : graphs (the usual forests and trees). Theorem (easy greedy algorithm) Let T be a (partial) k-forest of v vertices. Then ( ) n ex (k) (n,t) (v k). k 1 Gives the correct order of magnitude. If T =, ex ( n 1 k 1).

8 A partial forest is not necessarily a forest t+1 0 t t t is not a forest! It is a partial k-tree for k 3. C (k) t t t 1 2t 1

9 A starting point: The Erdős Ko Rado theorem The simplest forest possible: disjoint hyperedges. F ( [n]) k, a k-uniform set-system on the n-element underlying set [n] = {1,2,...,n}, n k 2. Theorem (Erdős Ko Rado 1961) If hyperedges pairwise intersect each other and n 2k, then ( ) n 1 F. k 1 EQUALITY only for centered systems (n > 2k). Construction: all k-element subsets containing a given element F 1 := {F : F = k, 1 F [n]}.

10 Erdős Matching Conjecture matching number: ν(f) := the maximum number of pairwise disjoint members (edges) of F. M (k) ν := ν pairwise disjoint k-sets. A (k) (ν) := ( [kν 1]) ( k, A = kν 1 ) k. B n (k) (ν) := {B ( [n]) ( : B [ν 1] }, B = n ( k) n ν+1 ) k k Conjecture (The Matching Conjecture, Erdős 1965) If F ( [n]) k, ν(f) < ν and n kν 1 then {( kν 1 F ex (k) (n,m ν ) = max k ), The right hand side is (ν 1+o(1)) ( n 1 k 1). ( ) n k ( )} n ν + 1 k

11 The status of Erdős Matching Conjecture Obvious for k = 1, (singletons) True for graphs k = 2 (Erdős, Gallai, 1959) The case ν = 2 is the classical ( ) Erdős, Ko, Rado n 1 ex (k) (n,m ν ) (ν 1) ( n, k, ν) by Frankl k 1 True for n > n 0 (k,ν). Erdős n 0 (k,ν) < 2k 3 ν (Bollobás, Daykin, Erdős, 1976) n 0 (k,ν) < O(kν 2 ) (Frankl & ZF, unpublished) More recently ( ) Alon, Frankl, Huang, Rödl, Ruciński, and Sudakov Alon, Huang and Sudakov/ Frankl, Rödl and Ruciński: connections to other problems n 0 (k,ν) < 3k 2 ν (Huang, Loh and Sudakov, 2011) k = 3 SOLVED Łuczak and Mieczkowska for large ν, Frankl ν Best bound (Frankl 2012): True for n (2ν 1)k ν.

12 Tight k-uniform trees Definition (tight k-tree) A sequence of k-element sets T := {E 1,E 2,...,E u } such that E i (2 i u) there exists an earlier edge E α E i = k 1, α = α(i) < i and E i has a new vertex: E i \ j<i E j = 1. For k = 2 (i.e., for graph) this is the usual tree. ex (k) (n,t) =? when T (and k) are fixed and n.

13 The Turán number of tight k-trees Let T be a tight k-tree of v vertices. One can obtain a T-free k-graph by packing (v 1)-vertex sets: Two sets overlap in k 2. Replace them by complete k-graphs. Rödl, 1985, gives ex (k) (n,t) (1 o(1)) ( n k 1 ( v 1 k 1 ) ( ) v 1 ) k = (1+o(1)) v k ( ) n. k k 1 By a recent result of Keevash 2014: Given T, for many n s there is no need for (1+o(1)). The error term is only O(n k 2 ). Conjecture (Erdős and Sós for graphs, Kalai 1984 for all k) T ex (k) (n,t) v k ( ) n. k k 1

14 Starlike trees The Kalai conjecture has been proved (Frankl and ZF, 1987) for star-like tight trees, i.e., when T has a central edge meeting all others in k 1 vertices. For k = 2 these are the diameter 3 trees.

15 Other forests? ex(n,t) = Θ(n k 1 ) was simple. (For T =.) We are looking for exact formulas for ( ) n lim n ex(k) (n,t)/. k 1 We have seen examples where the extremal family was spreading out evenly as a design (starlike tight trees). For some other forests the extremal families were concentrated (Erdos thm for ν disjoint k-sets). Aim of this lecture: New exact asymptotic for ex(n,t) for a large class of trees with Erdős type, concentrated extremum.

16 Hypergraph expansions Definition (k-expansion) Given a (hyper)graph G the k-expansion of G is the k-graph G (k) obtained from G by enlarging each edge E of G with a set of k E vertices disjoint from V(G) such that distinct edges are enlarged by disjoint (k E )-sets. Example: Given a (usual) tree T. T = T (k) (the k-expansion). =

17 Graph expansions Expansions of a matching = Erdős theorem. Theorem (Frankl 1977, a path of two edges. Conj d by Erdős-Sós) Fix k 4. Then ex k (n,p (k) 2 ) = ( n 2 k 2) for n > n0 (k), with equality only for a 2-star (two points are contained in all k-sets). The case k = 4 was completed for all n by Keevash, Mubayi, Wilson 06. The case χ(g) > k by Pikhurko / and by Mubayi Then ex k (n,g (k) ) = Θ(n k ).

18 Expanded triangles (k) := {A,B,C}, A B = B C = C A = 1 and A B C =. Theorem (Frankl&ZF 1987 Chvátal s 1972 triangle conjecture) If F ( [n]) k does not contain a (k) (and n > n k, k 3), then F ( n 1 k 1). Equality only for centered families. Does not imply EKR.

19 Linear paths A 5-edge linear (loose, 1-tight) path, P (k) 5. Cannot be covered by two vertices. ex(n,p (k) 5 ) F 2 = ( n 1 k 1 ) + ( ) n 2. k 1 F t := {F ( [n]) k : F {1,2,...,t} }. F t does not contain a linear path of length 2t + 1, P 2t+1.

20 The Turán number of a linear path Theorem (ZF, T. Jiang, R. Seiver 2011, t = 1 Frankl, ZF 1987) If F ( [n]) k does not contain a linear path of length 2t + 1, k 4 and n > n k,t then F F t. ( ) ( ) n 1 n t ex(n,p 2t+1 ) = + +. k 1 k 1 Equality only for F t. For large n improves Győri, Katona, Lemons 2010 by a factor of k, and Mubayi, Verstraëte 2007 by a factor of 2. Case k = 3: same true. By Kostochka, Mubayi, Verstraëte

21 The Turán number of a linear path, even case For even length: one can add a 2-intersecting system to F t. Theorem (same authors 2011, t = 0 Frankl 1977) If n > n k,t (we have kt < O(log log n)) and k 4, then ex(n,p 2t+2 ) = ( ) n k 1 ( ) n t k 1 ( ) n t 2 +. k 2 The only extremal system: F t {F : F = k,{t + 1,t + 2} F {t + 1,...,n 1,n}}. Proof: By Delta-system method! Again, the case k = 3 was completed by Kostochka, Mubayi, Verstraëte Different method, for very large n.

22 The Turán number of a linear cycles Theorem (ZF, T. Jiang, 2013, t = 1 Frankl, ZF 1987) If F ( [n]) k does not contain a linear CYCLE of length 2t + 1 k 5 and n > n k,t then F F t. ( ) ( ) n 1 n t ex(n,c 2t+1 ) = + +. k 1 k 1 Equality only for F t. We also have an asymptotic for k = 4. ex = (t + o(1)) ( n 1 k 1). Kostochka, Mubayi and Verstraëte for k = 3. Same bound as for PATH, P 2t+1, but more difficult to prove. Neither result implies the other. The case of ex(n,c 2t+2 ) is a slightly more complicated.

23 Crosscuts in hypergraphs Definition (Frankl and ZF 1987) Given an k-graph H, a crosscut of H is subset X of V(H) such that H meets all members of H in a singleton. σ(h) := min X (it it exists). F 0 σ 1 Ft 0 := {F ( [n]) k : F {1,2,...,t} = 1}. does not contain H. ( ) n 1 ex (k) (n,h) F σ 1 = (σ(h) 1+o(1)). k 1

24 The Turán number of expanded forests = Theorem (Asymptotic for blown up linear forests (ZF 2011)) For k 4 we have ( ) ( ) n σ + 1 n 1 (σ 1) ex(n,t (k) ) = (σ 1+o(1)). k 1 k 1 Kostochka-Mubayi-Verstraëte 2014: previous results for expansions of paths, cycles and trees hold for all k 3 (esp for triple systems, k = 3).

25 Main result: expansions of (k 2)-forests Theorem (ZF & T. Jiang 2014+) Suppose that H is a k-forest such that each edge has at least two degree 1 vertices, σ := σ(h). Then ( ) ( ) n σ + 1 n 1 (σ 1) ex(n,h) = (σ 1+o(1)). k 1 k 1 Error term is only O(n k 2 ). Stability of extremum: yes. From this: exact results. Same for H H (partial forests like the cycle C (k) l ), σ := σ(h ). It does not hold with only one degree 1 vertex on each edge. Except for k = 3 (Kostochka et al.)

26 Some tools of the proof 1. Delta system, consider kernels. 2. The structure of typical hyperedges (Type I). 3. Separating large degree vertices, L. 4. Typical k-sets meet L in a singleton. 5. The number of typical edges (σ 1) ( n l k 1). 6. To prove a Kruskal-Katona type thm for the shadow of H-free hypergraphs. 7. A weak stability of the extremum.

27 Stability of ext(n, K p+1 ) Theorem (Large p-chromatic subgraphs (ZF 2010)) Suppose K p+1 G, V(G) = n and e(g) e(t n,p ) t. Then there exists a p-chromatic subgraph H 0, E(H 0 ) E(G) such that e(h 0 ) e(g) t. There are other (more exact) results, (Hanson, Toft 1991, Győri 1987, 1991, Alon 1996, Tuza et al., Bollobás & Scott,...) But! here there is no ε,δ,n 0,.... It is true for every n and t. Corollary (ZF 2010, a special case of Simonovits Stability thm.) Suppose G is K p+1 -free with e(g) e(t n,p ) t. Then a complete p-chromatic graph H, V(H) = V(G), such that E(G) E(H) 3t.

28 The superstability of Erdős theorem Theorem (Recall Erdős Matching theorem) If F ( [n]) k, ν(f) < ν and n > n0 (k,ν), then ( ) ( ) ( ) n n ν + 1 n 1 F ex (k) (n,m ν ) = = (ν 1) +O(n k 2 ). k k k 1 Much more is true. It is easy to prove a concentration theorem. F: ν(f) < ν implies that a set L of size ν 1 such that {F F : F L = } = O(n k 2 ). All but a small number of hyperedges meet L.

29 No superstability for most trees Construction of H(m,r,t). r t 2, α Z, V(H) := [r] [m]. F = {(1,y 1 ),(2,y 2 ),...,(r,y r )} H if and only if 1 i t y i α mod m. H = (n/r) r 1 and L < n/(2r) misses at least 1 2 (n r )r 1 edges. So if T has a superstability property (like M ν does), then it must be embedded into H(m,r,t) for all t 2 for all large m. Theorem (Main step in the proof of our results) If T can be embedded into all H(m,r,t), and every E T has a degree 1 vertex, then L, L = n ε, such that {F F : F L = } = O(n k 1 ε ). Almost all but a small number of hyperedges meet L.

30 A powerful tool: delta systems Def: The sets {D 1,D 2,...,D l } are forming a delta-system of size l with center (kernel) C if D i D j = C ( 1 i < j l). C D 1 D 2 D 3 D 4 D 5 D 6

31 The Erdős-Rado bound for -systems Theorem (Erdős, Rado 1960, large -systems in k-unif families) ϕ(k,l): If F > ϕ(k,l), then it contains -system of size l. (l 1) k ϕ(k,l) < k!(l 1) k. ( ) k (Kostochka 1997): ϕ(k, 3) < Ck! log log log k log log k Erdős: ϕ(k,3) < C k? ($1,000).

32 k-partite hypergraphs The intersection structure of F F with respect to the family F I(F,F) := {F F : F F,F F }. Def: a k-uniform family F ( [n]) k is k-partite if a k-partition [n] = X 1 X k such that F X i = 1 for i and F F. Then a natural projection π : 2 [n] [k]. π(s) := {i : S X i } & π(i(f,f)) := {π(s) : S I(F,F)}. The -system method Algebraic methods frequently do not yield sufficient information on local structure. We seek new methods, building blocks.

33 The intersection semilattice lemma The next statement is a kind of compactness. Instead of the whole F ( [n]) k one should consider only a J 2 [k], where the kernel of kernels is again a kernel. Theorem (ZF 1983) There exists a positive bound c(k,l) > 0 as follows: Every k-uniform family F contains a subsystem F F such that (1) F c(k,l) F, (2) F k-partite, (3) the intersection structure is homogeneous, J 2 {1,2,...,k} such that π(i(f,f )) = J for all edge F F, (4) J closed under intersection, (A,B J = A B J ), (5) every pairwise intersection is a kernel, i.e., F 1,F 2 F F 3,...,F l F such that {F 1,F 2,...,F l } is a -system. The End