Near-sunflowers and focal families

TL;DR

This paper investigates variants of sunflowers in set families, improving bounds on binary vectors with specific splitting properties and proposing a weaker form of the Erdős-Rado sunflower conjecture.

Contribution

It provides new bounds on binary vectors with splitting properties and introduces a weaker version of the Erdős-Rado sunflower conjecture.

Findings

Improved an upper bound on the number of binary vectors with a splitting property.

Proposed a weaker version of the Erdős-Rado sunflower conjecture.

Analyzed variants of sunflowers in set families.

Abstract

We present some problems and results about variants of sunflowers in families of sets. In particular, we improve an upper bound of the first author, K\"orner and Monti on the maximum number of binary vectors of length $n$ so that every four of them are split into two pairs by some coordinate. We also propose a weaker version of the Erd\H{o}s-Rado sunflower conjecture.