Focal-free uniform hypergraphs and codes

TL;DR

This paper explores the relationship between focal-free hypergraphs and the Erdős Matching Conjecture, providing asymptotically optimal bounds and explicit sizes for maximum focal-free hypergraphs and codes.

Contribution

It establishes a connection between focal-free hypergraphs and the Erdős Matching Conjecture, improving bounds and explicitly determining sizes for various parameters.

Findings

Asymptotically optimal bounds on hypergraph sizes

Explicit sizes for hypergraphs and codes with many parameters

Significant improvement over previous results

Abstract

Motivated by the study of a variant of sunflowers, Alon and Holzman recently introduced focal-free hypergraphs. In this paper, we show that there is an interesting connection between the maximum size of focal-free hypergraphs and the renowned Erd\H{o}s Matching Conjecture on the maximum number of edges that can be contained in a uniform hypergraph with bounded matching number. As a consequence, we give asymptotically optimal bounds on the maximum sizes of focal-free uniform hypergraphs and codes, thereby significantly improving the previous results of Alon and Holzman. Moreover, by using the existentce results of combinatorial designs and orthogonal arrays, we are able to explicitly determine the exact sizes of maximum focal-free uniform hypergraphs and codes for a wide range of parameters.