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The Exact Erdős-Ko-Rado Theorem for 3-wise $t$-intersecting uniform families

Peter Frankl, Jian Wang

Abstract

Let $\mathcal{F}$ be a family of $k$-element subsets of $\{1,2,\ldots,n\}$. For $t\geq 1$, we say that $\mathcal{F}$ is {\it 3-wise $t$-intersecting} if $|F_1\cap F_2\cap F_3|\geq t$ for all $F_1,F_2,F_3\in \mathcal{F}$. In the present paper, we prove that if $\mathcal{F}$ is 3-wise $t$-intersecting and $n\geq \frac{\sqrt{4t+9}-1}{2}k$, $k>t\geq 46$, then $|\mathcal{F}|\leq \binom{n-t}{k-t}$. The restriction on $n$ is asymptotically best possible. The corresponding result for non-trivial 3-wise $t$-intersecting families is obtained as well for $n\geq \frac{\sqrt{4t+9}-1}{2}k$ and $k>t\geq 55$.

The Exact Erdős-Ko-Rado Theorem for 3-wise $t$-intersecting uniform families

Abstract

Let be a family of -element subsets of . For , we say that is {\it 3-wise -intersecting} if for all . In the present paper, we prove that if is 3-wise -intersecting and , , then . The restriction on is asymptotically best possible. The corresponding result for non-trivial 3-wise -intersecting families is obtained as well for and .
Paper Structure (6 sections, 30 theorems, 158 equations)

This paper contains 6 sections, 30 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Tools of the Proofs
  3. Some structural lemmas
  4. Proof of Theorem \ref{['thm-main']}
  5. Proof of Theorem \ref{['thm-main2']}
  6. Concluding remarks

Key Result

Theorem 1.2

Suppose that $\mathcal{F}\subset \binom{[n]}{k}$ is 2-wise $t$-intersecting, $k\geq t\geq 1$, $n\geq (t+1)(k-t+1)$. Then

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1.2: Exact Erdős-Ko-Rado Theorem, EKR, F78, W84
  • Theorem 1.3: FF91, AK0
  • Theorem 1.4: HM, F78-2, AK
  • Corollary 1.6
  • Theorem 1.7: BL
  • Theorem 1.8: FW25
  • Theorem 1.9: FW23
  • Theorem 1.10
  • Theorem 1.11
  • ...and 38 more