Linear dependencies, polynomial factors in the Duke--Erd\H os forbidden sunflower problem
Andrey Kupavskii, Fedor Noskov
TL;DR
This work advances the forbidden sunflower problem by proving near-optimal upper bounds for k-uniform families on [n] that avoid sunflowers with s petals and a core of size t-1, in regimes where n grows polynomially with k. It unifies three powerful methods—the Delta-system (sunflower) technique, spread-approximation, and Boolean-hypercontractivity—to handle broad parameter ranges and to extend results to spread domains, including permutations and product-domain families. The central contributions include a polynomial-dependency main theorem, a general spread-domain no-sunflower theorem with explicit remainder terms, and parallel results for subfamilies of permutations and product-domain structures, together with a comprehensive treatment of delta-systems, sunflower Turán-type bounds, and sharp-threshold machinery. The findings significantly broaden the Delta-system method toolkit, offering a cohesive framework that yields polynomial dependencies and cross-domain applicability, with potential implications for related Turán-type problems and designs.
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 $φ(s,k)$ 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 Füredi 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 Füredi 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.