Fractional decompositions of dense hypergraphs

1. Fractional decompositions of dense hypergraphs Raphael YusterUniversity of Haifa

2. Definitions, notations and background The Rödl nibble The edges of the complete r-graph K(n,r) can be packed, almost completely, with copies of K(k,r), where k is fixed. This result is considered one of the most fruitful applications of the probabilistic method. It was not known whether the same result also holds in a dense hypergraph setting.

3. Example: Packing 7 copies of K(3,2) in a K(7,2)

4. Definitions, notations and background Let H0 be a fixed hypergraph.A fractional H0-decomposition of a hypergraph H is an assignment of nonnegative real weights to the copies of H0 in H such that for each e E(H) the sum of the weights of copies of H0containing e is 1. Example:K4 has a fractional K3-decomposition. Each triangle receives a weight of ½.

5. Main result There exists a positive constant α=α(k,r) so that everyn-vertex r-graph in which every (r-1)-set is contained in at least (n-r+1)(1-α) edges has a fractionalK(k,r)-decomposition. • The proof is algorithmic. • K(n,r) is simply the case α=0 since every r-1 set is contained in n-r+1 edges. • In fact we obtain: α(k,r) > 6-krα(k,2) > 0.1k -10α(3,2) > 10-4

6. From fractional to integral • Combined with the following result of Rödl, Schacht, Siggers and Tokushige our result has consequences for integral packing. • Let ν(H0,H) denote the maximum number of edge-disjoint copies of H0 in H(the H0-packing number of H). • Let ν*(H0,H) denote the fractional relaxation. • Trivially, ν*(H0,H) ≥ ν(H0,H). • They proved that if H is an r-graph with n vertices then ν*(H0,H) < ν(H0,H) + o(nr).

7. Corollaries for graphs • If G is a graph with n vertices, andδ(G) > (1- 1/10k10)n then G has an asymptotically optimal Kk-packing. • Same theorem holds for k-vertex graphs. • For triangles (k=3), δ(G) > 0.9999n suffices. • The previously best known bound (for the missing degree) in the triangles case was 10-24 (Gustavsson).The previously best known bound for Kk was 10-37k-94(Gustavsson).

8. Corollary for 3-graphs • If H is a 3-graph with n vertices andminimum co-degree (1-216-k)n then H has an asymptotically optimal K(k,3)-packing. • Same theorem holds for k-vertex 3-graphs. • The previously best known bound (for the missing co-degree) was 0 (The Rödl nibble).

9. Tools used in the proof • Some linear algebra. • Kahn’s Theorem:For every r* > 1 and every γ > 0 there exists a positive constant ρ=ρ(r*,γ) such that the following statement is true: If U is an r*-graph with: (i) maxdeg < D (ii) maxcodeg < ρD then there is a proper coloring of the edges of U with at most (1+γ)D colors.

10. Tools used in the proof • Several probabilistic arguments large deviation, local lemma. • Hall’s Theorem for hypergraphs (by Aharoni and Haxell. Has a topological proof): Let U={U1,…,Um} be a family ofp-graphs. If for every W  U there is a matching in UU  WU of size greater than p(|W|-1) then U has an SDR.

11. The proof Recall our goal: There exists a positive constant α=α(k,r) so that every r-graph in which every (r-1)-set is contained in at least n(1-α) edges has a fractional K(k,r)-decomposition. • Let t=k(r+1) and consider the family of the 3 r-graphs:F(k,r) = { K(k,r), K(t,r), H(t,r) }where H(t,r) is a K(t,r) missing one edge. • Lemma: K(k,r) fractionally decomposes each element of F(k,r). (To show that K(k,r) fractionally decomposes H(t,r) requires some work. We use linear algebra here.)

12. The proof For r=2 (graphs) it suffices to take t=2k-1 and the lemma is easy. Example: r=2, k=3 hencet=5 and F(3,2) = { K(3,2), K(5,2), H(5,2) } 1 2 w(i,x,y)=1/2 i=1,2 x,y=a,b,c H(5,2) c a b

13. The proof It suffices to prove the stronger theorem There exists a positive constant α=α(k,r) so that everyr-graph in which every (r-1)-set is contained in at leastn(1-α) edges has an integralF(k,r)-decomposition. • Let ε = ε(k,r) be chosen later. • Let η = (2-H(ε)0.9)1/ε. H(ε) the entropy function. • Let α = min{ (η/2)2 , ε2/(t24t+1)} • Let γ satisfy (1-αt2t)(1-γ)/(1+γ)2 > 1-2αt2t • Let r* = • Let ρ = ρ(r*,γ) be the constant from Kahn’s theorem. • Assume nis suff. large as a function of all these constants.

14. The proof • Let δd(H) and Δd(H) denote the min and maxd-degrees of H, 0 < d < r. • Our r-graph H satisfies δd(H) > • It is not difficult to prove (induction) that every edge of H lies on many K(t,r). In fact, if c(e) denotes the number of K(t,r) containing e then

15. The proof • Color the edges of Hrandomly usingq=n1/(4r*-4) colors (that’s many colors!) • Let Hi be the spanning r-graph colored with i. • Easy (Chernoff):δd(Hi) very close to δd(H)/q • Not so easy: we would also like to show thatci(e) is very close to its expectation c(e)n-1/4.Note that two K(t,r) that contain e may share other edges as well – a lot of dependence.

16. The proof • By considering the dependency graph of the c(e) events we can show:(1+γ)nt-r-1/4> ci(e) (t-r)! > (1-γ)nt-r-1/4(1-αt2t) • We fix the coloring with qcolors satisfying the above. • For each Hi we create another r*-graph Ui:- the vertices of Ui are the edges of Hi- the edges of Uiare the copies of K(t,r) in Hi • Notice that Δ(Ui) < D = (1+γ)nt-r-1/4(t-r)!-1Notice that Δ2(Ui) < nt-r-1 << ρD

17. The proof • By Kahn’s theorem this means that the K(t,r) copies of Hican be partitioned into at most D(1+γ) packings. • We pick one of these packings at random. Denote it Li. • The set L=L1U…ULqis a K(t,r) packing of H. • Let M denote the edges of H not belonging to any element of L. • Let p = A p-subset {S1,…,Sp} of L is good fore Mif we can select one edge from each Sisuch that, together with p, we have a K(k,r). • We say that L is good if for each e M we can select a good p-subset, and all |M| selections are disjoint.

18. Example: being good k=3 r=2 So: t=5 p=2 b a S100 a c M L b S700 c {S100,S700} is good for (a,c)

19. The proof • Recall F(k,r) = { K(k,r), K(t,r), H(t,r) }.Clearly:Lis good → H has an Fk-decomposition. • It remains to show that there exists a good L. We will show that with positive probability, the random selection of the qpackings L1U…ULq yields a good L. • We use Hall’s theorem for hypergraphs.

20. The proof • Let M={e1,…,em}. • Let U={U1,…,Um} be a family of p-graphs defined as follows: • The vertex set of Uiis L (i.e, K(t,r) copies) • The edges of Ui are the p-subsets of L good for ei • U has an SDR → L is good. • Thus, it suffices to show that the random selection of the q packings L1U…ULq guarantees that, with positive probability, for everyW  U there is a matching in UU  WU of size greater than p(|W|-1).

21. The proof • It turns out that the only thing needed to guarantee this is to show that with positive probability, for all 1 < d < r:Δd(H[M]) < 2ε • Once this is established, the remainder of the claim is deterministic, namely • Δd(H[M]) < 2ε →U has an SDR.(purely combinatorial proof, but not so easy).

22. Open problems • Determine the correct value of α(k,r). • The simplest case is α(3,2) (triangles). We currently have α(3,2) > 10-4. • A construction shows that α(3,2) ≤ ¼. • More generally, a construction given in the paper shows that α(k,2) ≤ 1/(k+1).We conjecture α(k,2) = 1/(k+1). • For hypergraphs we don’t even know what to conjecture.