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_n^r into approximate high girth (n,q,r)-Steiner systems. If time permits, I will explain some of the other applications of our main results such as to rainbow matchings. This is joint work with Luke Postle.
05/17/2022 9:30 am - 10:30 am