Finding an almost perfect matching in a hypergraph avoiding forbidden submatchings

Abstract

In 1973, Erdős conjectured the existence of high girth (n, 3, 2)-Steiner systems. Recently, Glock, Kühn, Lo, and Osthus and independently Bohman and Warnke proved the approximate version of Erdős' conjecture. Just this year, Kwan, Sah, Sawhney, and Simkin proved Erdős' conjecture. As for Steiner systems with more general parameters, Glock, Kühn, Lo, and Osthus conjectured the existence of high girth (n, q, r)-Steiner systems. We prove the approximate version of their conjecture.This result follows from our general main results which concern finding perfect or almost perfect matchings in a hypergraph G avoiding a given set of submatchings (which we view as a hypergraph H where V (H) = E(G)). Our first main result is a common generalization of the classical theorems of Pippenger (for finding an almost perfect matching) and Ajtai, Komlós, Pintz, Spencer, and Szemerédi (for finding an independent set in girth five hypergraphs). More generally, we prove this for coloring and even list coloring, and also generalize this further to when H is a hypergraph with small codegrees (for which high girth designs is a specific instance). Indeed, the coloring version of our result even yields an almost partition of K r n into approximate high girth (n, q, r)-Steiner systems. Our main results also imply the existence of a perfect matching in a bipartite hypergraph where the parts have slightly unbalanced degrees. This has a number of other applications; for example, it proves the existence of ∆ pairwise disjoint list colorings in the setting of Kahn's theorem for list coloring the edges of a hypergraph; it also proves asymptotic versions of various rainbow matching results in the sparse setting (where the number of times a color appears could be much smaller than the number of colors) and even the existence of many pairwise disjoint rainbow matchings in such circumstances.

Conflict-free hypergraph matchings

“…Note added. While finalising our paper, we learned that Delcourt and Postle [8] independently obtained similar results. In particular, they were also motivated by and proved the existence of approximate high-girth Steiner systems, and observed that this is just a special case of a general hypergraph matching theorem.…”

“…of this application and in our case we need polylogarithmic degree). Our result has the advantage that we can track test functions (which we believe is crucial for potential applications based on the absorption method) and handle conflicts of size 2 without requiring a girth condition for C (see Theorem 1.16 in [8]).…”

On the $(6,4)$-problem of Brown, Erdős and Sós

“…In this section, we introduce conflict-free hypergraph matchings, a general tool developed recently by Glock, Joos, Kim, Kühn and Lichev [9], and independently by Delcourt and Postle [4]. These works were motivated in turn by earlier results of Glock, Kühn, Lo and Osthus [10] as well as Bohman and Warnke [2], who proved an approximate version of an old conjecture of Erdős [6] on the existence of high-girth Steiner triple systems.…”

“…Hence, the above lower bound has the correct order of magnitude. If not, then even determining the order of magnitude of f (r) (n; s, k) is a major open problem, which encompasses for instance the famous (7,4)-problem and its generalisations (see e.g. [1]).…”

“…The challenge is to ensure that in this packing, we do not create any (6, 4)-configurations using edges from different copies of T 7 . For this, a crucial ingredient in our proof is a recent result on "conflict-free hypergraph matchings" [4,9], which we introduce in Section 2.…”