Ryser's Conjecture for $t$-intersecting hypergraphs

Abstract

A well-known conjecture, often attributed to Ryser, states that the cover number of an r-partite r-uniform hypergraph is at most r − 1 times larger than its matching number. Despite considerable effort, particularly in the intersecting case, this conjecture remains wide open, motivating the pursuit of variants of the original conjecture. Recently, Bustamante and Stein and, independently, Király and Tóthmérész considered the problem under the assumption that the hypergraph is tintersecting, conjecturing that the cover number τ (H) of such a hypergraph H is at most r − t. In these papers, it was proven that the conjecture is true for r ≤ 4t − 1, but also that it need not be sharp; when r = 5 and t = 2, one has τ (H) ≤ 2.We extend these results in two directions. First, for all t ≥ 2 and r ≤ 3t − 1, we prove a tight upper bound on the cover number of these hypergraphs, showing that they in fact satisfy τ (H) ≤ ⌊(r − t)/2⌋ + 1. Second, we extend the range of t for which the conjecture is known to be true, showing that it holds for all r ≤ 36 7 t − 5. We also introduce several related variations on this theme. As a consequence of our tight bounds, we resolve the problem for k-wise t-intersecting hypergraphs, for all k ≥ 3 and t ≥ 1. We further give bounds on the cover numbers of strictly t-intersecting hypergraphs and the s-cover numbers of t-intersecting hypergraphs.

Generalizations and strengthenings of Ryser's conjecture

“…The best known results are due to Bishnoi, Das, Morris, Szabó [12]; in fact, when t > r 3 , their results are tight. Theorem 8.5 (Bishnoi, Das, Morris, Szabó [12]).…”

“…The best known results are due to Bishnoi, Das, Morris, Szabó [12]; in fact, when t > r 3 , their results are tight. Theorem 8.5 (Bishnoi, Das, Morris, Szabó [12]). Let t and r be integers with r 3 < t ≤ r. If H is an r-partite hypergraph in which every pair of edges has intersection size at least t, then τ (H) ≤ r−t+1 2 , and this is best possible.…”

“…Bishnoi, Das, Morris, Szabó [12] also studied a generalization where every set of k edges intersect in at least t vertices. It turns out that their result is a generalization of Theorem 7.1 (which is the case k ≥ 3, t = 1 below).…”

“…It turns out that their result is a generalization of Theorem 7.1 (which is the case k ≥ 3, t = 1 below). Theorem 8.6 (Bishnoi, Das, Morris, Szabó [12]). Let k, r, and t be integers with k ≥ 3, r ≥ 2, and t ≥ 1.…”