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Exact results and the structure of extremal families for the Duke--Erdős forbidden sunflower problem

Andrey Kupavskii, Fedor Noskov

TL;DR

This work resolves the Duke–Erdős forbidden sunflower problem in the regime $t=2$, odd $s$, and large $n$, delivering an exact extremal construction and a stability theory. The authors develop a first-order stability result for the problem and reveal a near-reduction to a low-dimensional Erdős–Rado-type problem that depends only on $t$ and $s$. They then establish an explicit extremal structure: for odd $s$, the extremal $k$-uniform family avoiding sunflowers with core size $1$ is captured by a two-clique template, with a precise containment criterion and a unique (up to isomorphism) extremal form. The graph-case and exact Erdos–Duke results are deduced by iterating the stability framework to full extent, yielding a comprehensive, structure-based resolution of the Duke–Erdős problem in this parameter range.

Abstract

In 1977, Duke and Erdős asked the following general question: What is the largest size of a family $\mathtt{F} \subset \binom{[n]}{k}$ that does not contain a sunflower with $s$ petals and core of size exactly $t - 1$? This problem is closely related to the famous Erdős--Rado sunflower problem of determining the size $φ(s,t)$ of the largest $t$-uniform family with no $s$-sunflower. In this paper, we answer this question exactly for $t=2$, odd $s$ and $k\ge 5$, provided $n$ is large enough. Previously, the only know exact extremal result on this problem was due to Chung and Frankl from 1987. One of the important ingredients for the proof that we obtained is a stability result for the Duke--Erdős problem, which was previously not known, mostly due to our lack of understanding of the behaviour of $φ(s,t)$. For large $k$ and $n$ we in fact manage to reduce the Duke--Erdős problem to an Erdős--Rado-like problem which depends on $t$ and $s$ only. In particular, we get a good understanding of the structure of extremal families for the Duke--Erdős problem in terms of the Erdős--Rado problem. Previously, a much looser variant of this connection (only in terms of the sizes, rather than the structure, of respective extremal families) was established in a seminal work of Frankl and Füredi from 1987.

Exact results and the structure of extremal families for the Duke--Erdős forbidden sunflower problem

TL;DR

This work resolves the Duke–Erdős forbidden sunflower problem in the regime , odd , and large , delivering an exact extremal construction and a stability theory. The authors develop a first-order stability result for the problem and reveal a near-reduction to a low-dimensional Erdős–Rado-type problem that depends only on and . They then establish an explicit extremal structure: for odd , the extremal -uniform family avoiding sunflowers with core size is captured by a two-clique template, with a precise containment criterion and a unique (up to isomorphism) extremal form. The graph-case and exact Erdos–Duke results are deduced by iterating the stability framework to full extent, yielding a comprehensive, structure-based resolution of the Duke–Erdős problem in this parameter range.

Abstract

In 1977, Duke and Erdős asked the following general question: What is the largest size of a family that does not contain a sunflower with petals and core of size exactly ? This problem is closely related to the famous Erdős--Rado sunflower problem of determining the size of the largest -uniform family with no -sunflower. In this paper, we answer this question exactly for , odd and , provided is large enough. Previously, the only know exact extremal result on this problem was due to Chung and Frankl from 1987. One of the important ingredients for the proof that we obtained is a stability result for the Duke--Erdős problem, which was previously not known, mostly due to our lack of understanding of the behaviour of . For large and we in fact manage to reduce the Duke--Erdős problem to an Erdős--Rado-like problem which depends on and only. In particular, we get a good understanding of the structure of extremal families for the Duke--Erdős problem in terms of the Erdős--Rado problem. Previously, a much looser variant of this connection (only in terms of the sizes, rather than the structure, of respective extremal families) was established in a seminal work of Frankl and Füredi from 1987.

Paper Structure

This paper contains 11 sections, 16 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Extremal families
  4. Tools
  5. Proof of Theorem \ref{['theorem: stability in Erdos-Duke']}
  6. Proof of Theorem \ref{['theorem: structural duke-erdos']}
  7. Proof of Proposition \ref{['proposition: bounds on same uniformity remainder']}
  8. Proof of Proposition \ref{['proposition: k-t-i fatness']}
  9. Proof of Proposition \ref{['proposition: cross matching']}
  10. Proof of Theorem \ref{['theorem: graph case Duke-Erdos extremal']}
  11. Proof of Theorem \ref{['theorem: exact erdos-duke']}

Key Result

Theorem 1.1

Let $s$ be odd and $k \geqslant 5$ and $n\geqslant n_0(s,k)$. Let $\mathcal{F} \subset \binom{[n]}{k}$ be an extremal family that does not contain a sunflower with $s$ petals and the core of size $1$. Then, there exists a graph $G = K_1 \sqcup K_2$ consisting of two disjoint cliques $K_1, K_2$ of si

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2: Frankl and Füredi, 1987
  • Example 1: Frankl1987
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Proposition 2.2
  • Lemma 3.1: Theorem 20.1.1 from spielman2019spectral
  • Theorem 3.2
  • Theorem 3.3: frankl1987shifting
  • ...and 18 more