Linear dependencies, polynomial factors in the Duke--Erd\H os forbidden sunflower problem

TL;DR

This paper advances the understanding of sunflower-free families by extending classical bounds to broader parameters using a unified approach that combines multiple advanced combinatorial techniques.

Contribution

It broadens the parameter range for sunflower-free families and unifies several methods in extremal set theory to achieve stronger results.

Findings

Extended bounds for sunflower-free families to polynomial functions of parameters.

Applied the unified method to various domains including permutations and simplicial complexes.

Achieved stronger results for families with forbidden sunflowers with small core sizes.

Abstract

We call a family of $s$ sets $\{F_1, \ldots, F_s\}$ a \textit{sunflower with $s$ petals} if, for any distinct $i, j \in [s]$, one has $F_i \cap F_j = \cap_{u = 1}^s F_u$. The set $C = \cap_{u = 1}^s F_u$ is called the {\it core} of the sunflower. It is a classical result of Erd\H os and Rado that there is a function $\phi(s,k)$ such that any family of $k$-element sets contains a sunflower with $s$ petals. In 1977, Duke and Erd\H os asked for the size of the largest family $\mathcal{F}\subset{[n]\choose k}$ that contains no sunflower with $s$ petals and core of size $t-1$. In 1987, Frankl and F\" uredi asymptotically solved this problem for $k\ge 2t+1$ and $n>n_0(s,k)$. This paper is one of the pinnacles of the so-called Delta-system method. In this paper, we extend the result of Frankl and F\"uredi to a much broader range of parameters: $n>f_0(s,t) k$ with $f_0(s,t)$ polynomial in $s$ and $t$. We also extend this result to other domains, such as $[n]^k$ and ${n\choose k/w}^w$ and obtain even stronger and more general results for forbidden sunflowers with core at most $t-1$ (including results for families of permutations and subfamilies of the $k$-th layer in a simplicial complex). The methods of the paper, among other things, combine the spread approximation technique, introduced by Zakharov and the first author, with the Delta-system approach of Frankl and F\"uredi and the hypercontractivity approach for global functions, developed by Keller, Lifshitz and coauthors. Previous works in extremal set theory relied on at most one of these methods. Creating such a unified approach was one of the goals for the paper.