Multicolor Sunflowers

TL;DR

This paper investigates bounds on the size of multiple families of subsets of [n] that avoid certain sunflower configurations, extending classical conjectures and providing sharp bounds for sum and product cases.

Contribution

It extends the sunflower conjecture to multiple families, establishing sharp bounds on their combined sizes when avoiding specific sunflower structures.

Findings

Maximum sum of families is $(k-1)2^n+1+inom{n}{n-k+2}+ o+inom{n}{n}$ for all $n \\geq k \\geq 3$.

For three families, the maximum product is between $(1/8+o(1))2^{3n}$ and $(0.13075+o(1))2^{3n}$.

Abstract

A sunflower is a collection of distinct sets such that the intersection of any two of them is the same as the common intersection $C$ of all of them, and $|C|$ is smaller than each of the sets. A longstanding conjecture due to Erd\H{o}s and Szemer\'edi states that the maximum size of a family of subsets of $[n]$ that contains no sunflower of fixed size $k>2$ is exponentially smaller than $2^n$ as $n\rightarrow\infty$. We consider this problem for multiple families. In particular, we obtain sharp or almost sharp bounds on the sum and product of $k$ families of subsets of $[n]$ that together contain no sunflower of size $k$ with one set from each family. For the sum, we prove that the maximum is $$(k-1)2^n+1+\sum_{s=n-k+2}^{n}\binom{n}{s}$$ for all $n \ge k \ge 3$, and for the $k=3$ case of the product, we prove that it is between $$\left(\frac{1}{8}+o(1)\right)2^{3n}\qquad \hbox{and} \qquad (0.13075+o(1))2^{3n}.$$