Tur\'{a}n numbers of sunflowers

TL;DR

This paper determines the asymptotic Turán number for hypergraph sunflowers with arbitrary uniformity, intersection size, and growing sunflower size, resolving a longstanding problem in extremal set theory.

Contribution

It provides the first asymptotic solution for the Turán number of sunflowers with arbitrary parameters, extending previous results limited to fixed sunflower sizes.

Findings

Established the asymptotic Turán number for sunflowers with arbitrary parameters.

Extended known results to cases where the sunflower size grows with the ground set.

Resolved a longstanding open problem in extremal set theory.

Abstract

A collection of distinct sets is called a sunflower if the intersection of any pair of sets equals the common intersection of all the sets. Sunflowers are fundamental objects in extremal set theory with relations and applications to many other areas of mathematics as well as theoretical computer science. A central problem in the area due to Erd\H{o}s and Rado from 1960 asks for the minimum number of sets of size $r$ needed to guarantee the existence of a sunflower of a given size. Despite a lot of recent attention including a polymath project and some amazing breakthroughs, even the asymptotic answer remains unknown. We study a related problem first posed by Duke and Erd\H{o}s in 1977 which requires that in addition the intersection size of the desired sunflower be fixed. This question is perhaps even more natural from a graph theoretic perspective since it asks for the Tur\'an number of a hypergraph made by the sunflower consisting of $k$ edges, each of size $r$ and with common intersection of size $t$. For a fixed size of the sunflower $k$, the order of magnitude of the answer has been determined by Frankl and F\"{u}redi. In the 1980's, with certain applications in mind, Chung, Erd\H{o}s and Graham and Chung and Erd\H{o}s considered what happens if one allows $k$, the size of the desired sunflower, to grow with the size of the ground set. In the three uniform case $r=3$ the correct dependence on the size of the sunflower has been determined by Duke and Erd\H{o}s and independently by Frankl and in the four uniform case by Buci\'{c}, Dragani\'{c}, Sudakov and Tran. We resolve this problem for any uniformity, by determining up to a constant factor the $n$-vertex Tur\'an number of a sunflower of arbitrary uniformity $r$, common intersection size $t$ and with the size of the sunflower $k$ allowed to grow with $n$.