Invariant covers of multipartite hypergraphs

TL;DR

This paper establishes a symmetric analogue of Lovász's estimate for the size of covers in multipartite hypergraphs, showing the existence of automorphism-invariant covers with bounded cardinality, extending previous results in hypergraph theory.

Contribution

It introduces a symmetric version of Lovász's estimate for hypergraph covers and generalizes existing bounds to automorphism-invariant covers in multipartite hypergraphs.

Findings

Existence of automorphism-invariant covers of size at most $nr/2$

Extension of Lovász's estimate to symmetric hypergraph covers

Generalization of previous bounds by Aharoni, Holzman, and Krivelevich

Abstract

We prove the following ``symmetric analogue'' of Lov\'asz's estimate (1975): if an $r$-partite hypergraph of rank $r\geqslant2$ has a cover of cardinality $n<\infty$, then it admits a cover of cardinality at most $nr/2$, which is invariant with respect to all automorphisms preserving the parts. We obtain also symmetric analogues of generalisations of Lov\'asz's estimate due to Aharoni, Holzman, and Krivelevich (1996).