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Multicolor Sunflowers

Dhruv Mubayi, Lujia Wang

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 Erdős and Szemerédi 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 $$\left(\frac{1}{8}+o(1)\right)2^{3n}\qquad \hbox{and} \qquad (0.13075+o(1))2^{3n}.$$

Multicolor Sunflowers

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 of all of them, and is smaller than each of the sets. A longstanding conjecture due to Erdős and Szemerédi states that the maximum size of a family of subsets of that contains no sunflower of fixed size is exponentially smaller than as . We consider this problem for multiple families. In particular, we obtain sharp or almost sharp bounds on the sum and product of families of subsets of that together contain no sunflower of size with one set from each family. For the sum, we prove that the maximum is for all , and for the case of the product, we prove that it is between

Paper Structure

This paper contains 9 sections, 7 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Sums
  4. Products
  5. Concluding remarks
  6. The Optimization Problem in the Proof of Theorem \ref{['thm3']}
  7. Case 1. $\mu_7=\mu_8=0$.
  8. Case 2. Exactly one of $\mu_7$ and $\mu_8$ is positive.
  9. Case 3. Both $\mu_7,\mu_8>0$.

Key Result

Theorem 1

There exists a constant $c$ such that if ${\cal A}\subset 2^{[n]}$ with $|{\cal A}|>2^{n-c\sqrt{n}}$ then $\cal A$ contains a sunflower with $3$ petals.

Theorems & Definitions (12)

  • Theorem 1: Erdős, Szemerédi ES78
  • Definition 2
  • Theorem 3
  • Theorem 4
  • Conjecture 1
  • Lemma 5
  • proof
  • Corollary 6
  • proof
  • Lemma 7
  • ...and 2 more