Piercing all maximum cliques in hypergraphs

TL;DR

This paper disproves a conjecture that large maximum cliques in hypergraphs can always be pierced by a fixed number of vertices, showing that no universal constant exists for this property.

Contribution

It provides a strong counterexample to the hypergraph clique piercing conjecture, demonstrating the absence of a universal threshold for the piercing number.

Findings

Counterexamples for all constants c<1, k≥3, t≥1

Maximum clique size can be large while the piercing number is unbounded

Refutes the possibility of a universal constant threshold for hypergraph clique piercing

Abstract

Graphs whose maximum clique size exceeds half of the total number of vertices satisfy a classical property: the family of their maximum sized cliques can be pierced by a single vertex. This result dates back to a 1965 theorem by Hajnal. Motivated by this theorem, Jung, Keszegh, P\'alv\"olgyi, and Yuditsky recently conjectured that an analogous result should hold for hypergraphs of larger uniformity, with an appropriate constant replacing the threshold $1/2$. In this paper we refute this conjecture in a strong form. We show that for any constant $c<1$ and integers $k\ge 3$ and $t\ge 1$, there exist $k$-uniform hypergraphs $G$ whose maximum clique size exceeds $c|V(G)|$, yet the family of maximum size cliques of $G$ cannot be pierced by $t$ vertices. This demonstrates that no universal constant threshold guarantees bounded piercing number for maximum cliques in uniform hypergraphs. We discuss further questions concerning the relationship between clique size and piercing maximum cliques in hypergraphs, and introduce a geometric variant of the problem using Helly's Theorem.