=q1(b1+ 1) +r1 with 0≤r1 ≤b1 and q1 ∼ n
clog2n withc:= 18003.
Ifr1 > n2log2n, we are done by settingb=b1. Otherwise, letb=b1− dcnlog4ne.Then n
3
=q1(b+ 1) + (r1+q1dcnlog4ne).
Moreover, for large enoughn, we obtain r1+q1dcnlog4ne< b, and therefore the remainder of the division n3
by (b+ 1) is at leastr1+q1dcnlog4ne> n2log2n, while 3000n2log2n≤ b≤3001n2log2n.
4.4 Concluding remarks
It is still an open problem what the threshold for the appearance of a Berge Hamilton cycle in randomr-uniform hypergraphs is. Theorem 4.1 implies an upper bound of log8rn/nr−1on this threshold, leaving a polylogarithmic gap to the best-known lower bound (see Theorem 1.9). It is worth mentioning that the proof of Theorem 4.1 works for slightly lower edge probabilities than log8rn/nr−1. However, using our proof method, the exponent of lognneeds to be strictly larger than a multiple of r. Since we do not believe that this is the correct answer, we did not intend to optimise the polylogarithmic factor for the sake of readability.
Having investigated the local resilience of random hypergraphs with respect to Berge Hamiltonicity, a natural problem to study next is the local resilience of random hypergraphs with respect to Hamilton cycles with a different notion of cycles, for instance with the notion of `-cycles. For tight Hamilton cycles such a result seems to be hard to achieve. Indeed, determining the local resilience of the complete r-uniform hypergraph with respect to tight
4.4. Concluding remarks 133
Hamilton cycles turned out to be challenging. The best-known bound on the minimum vertex degree of ann-vertex 3-uniform hypergraph that implies the containment of a tight Hamilton cycle is (5−√
5)/3 +ε n
2
, as shown by R¨odl and Ruci´nski [143]. Non-trivial bounds for higher uniformities are not known yet. R¨odl and Ruci´nski [142] posed the following conjecture, which would yield an asymptotically tight minimum vertex degree condition for tight Hamilton cycles.
Conjecture 4.32 (R¨odl, Ruci´nski [142]). For each integer r ≥3 and real ε >0 there is an integer n0 such that the following holds. If H is an r-uniform hypergraph on n≥n0 vertices with
δ1(H)≥ 1−
1−1 r
r−1 +ε
! n r−1
, then H contains a tight Hamilton cycle.
We remark that while the minimum vertex degree condition for weak Hamilton cycles in Proposition 4.23 is tight, the one that we established for Hamilton Berge cycles in Proposi-tion 4.24 can probably be improved.
Finally, using Theorem 4.1 and Theorem 4.25 we obtained a lower bound on the threshold bias of the (1 :b) Maker-Breaker game played onE(Kn(r)), where Maker wins if his hypergraph contains a Hamilton Berge cycle. In Subsection 4.3.2 we determined upper bounds on the threshold biases of Avoider-Enforcer games played on E(Kn(3)), where Avoider has to keep his hypergraph (almost) Berge-acyclic. We believe that for both games the bounds are not optimal and would be interested in knowing the threshold biases for the games that we studied.
Rainbow matchings in multigraphs 5
A conjecture by Aharoni and Berger suggests that every bipartite multigraph, the edges of which are coloured with n colours such that each colour class induces a matching of size n+ 1, contains a rainbow matching of size n. As elucidated in Subsection 1.2.2, this is a generalisation of famous open conjectures by Ryser and by Brualdi and Stein on Latin squares.
In this chapter we study general multigraphs, the edges of which are coloured with n colours. We prove that if each of thencolour classes covers 3n+o(n) vertices of the multigraph and induces a disjoint union of cliques, then there exists a rainbow matching of sizen. In the setting above, this implies that matching sizes of 3n/2 +o(n) suffice to guarantee a rainbow matching of size n. Thus our result marks a step towards the conjecture of Aharoni and Berger. Moreover, it solves an algebraic problem by Grinblat asymptotically.
In Section 5.1 we formulate our main result in terms of rainbow matchings in multigraphs, rainbow matchings in equivalence classes, and using the algebraic terminology of Grinblat’s question. We prove the equivalence of these formulations in Section 5.2. The proof of the main result is presented in Section 5.3. Finally, we close this chapter with a discussion of some open problems in Section 5.4.
This chapter is based on joint work with Dennis Clemens and Alexey Pokrovskiy [56] and on joint work with Dennis Clemens [53].
5.1 Introduction
In this chapter we asymptotically affirm a question by Grinblat on sets not belonging to alge-bras. Before formulating this question, let us recall thatv=v(n) was defined by Grinblat [85]
as the minimal cardinal number such that the following is true:
“LetA1, . . . ,Anbe algebras on a setXsuch that for eachi∈[n] there exist at least v(n) pairwise disjoint sets in P(X)\ Ai. Then there exists a family {Ui1, Ui2}i∈[n]
of 2npairwise disjoint subsets of X such that, for each i∈[n], if Q∈ P(X) and Q contains one of the two setsUi1 and Ui2 and its intersection with the other one is empty, then Q /∈ Ai.”
As shown by Grinblat [85], a lower bound onv(n) is 3n−2 for everyn∈N. He asked whether this bound is tight for everyn≥4:
135
Question 5.1 (Grinblat, [85]). Is it true that v(n) = 3n−2 for n≥4?
By proving the following theorem, we give an asymptotic answer to Question 5.1.
Theorem 5.2. For every δ >0 there exists n0=n0(δ) = 144/δ2 such that for every n≥n0 it holds that v(n)≤(3 +δ)n.
Nivasch and Omri [134] proved the upper boundv(n)≤16n/5 +O(1), using the following equivalent definition ofv(n) in the context of equivalence relations. LetX be a finite set and let A be an equivalence relation on X. Ifx, y∈X are equivalent underA, we writex∼Ay.
By
[x]A:={y∈X:x∼Ay}
we denote theequivalence class underAof an elementx∈X, while thekernel ofAis defined as
ker(A) :=
x∈X :|[x]A| ≥2 .
Following Nivasch and Omri [134], we letv1(n) be the minimal number such that ifA1, . . . , An are equivalence relations on X with |ker(Ai)| ≥ v1(n) for each i ∈ [n], then A1, . . . , An contain a rainbow matching, i.e. a set of 2n distinct elements x1, y1, . . . , xn, yn ∈ X with xi ∼Ai yi for each i∈ [n]. This identity is mainly based on the fact that there is a natural correspondence between algebras and equivalence relations. Using these definitions, it turns out that v(n) =v1(n) holds. Indeed, given an equivalence relation A on the set X, we can define the algebra A :=S
x∈S[x]A: S ⊆X . Conversely, given some algebra A on X, one can define the equivalence relation Aon X the equivalence classes of which are the inclusion minimal sets inA. A complete argument to show thatv(n) =v1(n) is presented in Section 5.2.
Thus, using the terminology of Nivasch and Omri [134] Theorem 5.2 is equivalent to the following theorem.
Theorem 5.3. For every δ > 0 there exists n0 = n0(δ) = 144/δ2 such that the following holds for every n ≥ n0. Let A1, . . . , An be n equivalence relations on a finite set X. If
|ker(Ai)| ≥(3 +δ)n for eachi∈[n], then A1, . . . , An contain a rainbow matching.
Observe that it would suffice to prove Theorem 5.3 for the case that each equivalence class ofA1, . . . , An has size 2 or 3. In the special case when all of these equivalence classes consist of 3 elements, the statement can be easily proved by a greedy argument even forδ = 0.
Theorem 5.3 can be rephrased in the context of graphs. If A1, . . . , An are equivalence relations on a set X, let the vertices of an edge-coloured multigraph be the elements of X and, for eachi∈[n], let{x, y} ∈ X2
be an edge of colouriif and only ifx∼Ai y. This means that the equivalence relations are represented in this multigraph by colour classes, each of which is the disjoint union of non-trivial cliques, i.e. complete graphs with at least 2 vertices.
A matching in an edge-coloured multigraph is called arainbow matching if all its edges have distinct colours. Using this notion, we can reformulate Theorem 5.3 as follows.
Theorem 5.4. For every δ > 0 there exists n0 = n0(δ) = 144/δ2 such that the following holds for everyn≥n0. LetGbe a multigraph, the edges of which are coloured withncolours.
If each subgraph ofGinduced by a colour class has at least(3 +δ)nvertices and is the disjoint union of non-trivial cliques, then Gcontains a rainbow matching of size n.