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Sunflowers and Ramsey problems for restricted intersections

Barnabás Janzer, Zhihan Jin, Benny Sudakov, Kewen Wu

TL;DR

The paper investigates Ramsey-type questions for set systems with restricted intersections, focusing on L-cliques and L-sunflowers and how large an L-intersection-free subfamily can be extracted when no L-sunflower of size m exists. It introduces color certificates and develops a refined delta-system framework to obtain near-optimal, single-exponential bounds in m and k, while contrasting these with the classical double-exponential Füredi lemma. In modular settings, it derives precise bounds depending on residues modulo p, including tight behavior for L = Z/pZ ackslash {0} and implications for quantum computing via shadow tomography and fractional chromatic numbers. The work provides algorithmic tools, notably a polynomial-time approach to construct large L-intersection-free subfamilies and to obtain fractional colorings, and it highlights open questions about optimal exponents and preserving sunflowers within subfamilies, informing both combinatorial theory and quantum applications.

Abstract

Extremal problems on set systems with restricted intersections have been an important part of combinatorics in the last 70 year. In this paper, we study the following Ramsey version of these problems. Given a set $L\subseteq \{0,\dots,k-1\}$ and a family $\mathcal{F}$ of $k$-element sets which does not contain a sunflower with $m$ petals whose kernel size is in $L$, how large a subfamily of $\mathcal{F}$ can we find in which no pair has intersection size in $L$? We give matching upper and lower bounds, determining the dependence on $m$ for all $k$ and $L$. This problem also finds applications in quantum computing. As an application of our techniques, we also obtain a variant of Füredi's celebrated semilattice lemma, which is a key tool in the powerful delta-system method. We prove that one cannot remove the double-exponential dependency on the uniformity in Füredi's result, however, we provide an alternative with significantly better, single-exponential dependency on the parameters, which is still strong enough for most applications of the delta-system method.

Sunflowers and Ramsey problems for restricted intersections

TL;DR

The paper investigates Ramsey-type questions for set systems with restricted intersections, focusing on L-cliques and L-sunflowers and how large an L-intersection-free subfamily can be extracted when no L-sunflower of size m exists. It introduces color certificates and develops a refined delta-system framework to obtain near-optimal, single-exponential bounds in m and k, while contrasting these with the classical double-exponential Füredi lemma. In modular settings, it derives precise bounds depending on residues modulo p, including tight behavior for L = Z/pZ ackslash {0} and implications for quantum computing via shadow tomography and fractional chromatic numbers. The work provides algorithmic tools, notably a polynomial-time approach to construct large L-intersection-free subfamilies and to obtain fractional colorings, and it highlights open questions about optimal exponents and preserving sunflowers within subfamilies, informing both combinatorial theory and quantum applications.

Abstract

Extremal problems on set systems with restricted intersections have been an important part of combinatorics in the last 70 year. In this paper, we study the following Ramsey version of these problems. Given a set and a family of -element sets which does not contain a sunflower with petals whose kernel size is in , how large a subfamily of can we find in which no pair has intersection size in ? We give matching upper and lower bounds, determining the dependence on for all and . This problem also finds applications in quantum computing. As an application of our techniques, we also obtain a variant of Füredi's celebrated semilattice lemma, which is a key tool in the powerful delta-system method. We prove that one cannot remove the double-exponential dependency on the uniformity in Füredi's result, however, we provide an alternative with significantly better, single-exponential dependency on the parameters, which is still strong enough for most applications of the delta-system method.

Paper Structure

This paper contains 16 sections, 24 theorems, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Forbidding L-cliques in the modular setting
  3. An efficient delta-system lemma
  4. Connections to quantum computing
  5. Colour certificates
  6. Forbidding L-sunflowers
  7. A new variant of the delta-system lemma
  8. Our new variant
  9. Double-exponential dependency in Füredi's delta-system lemma
  10. Set systems excluding one residue class
  11. Frankl--Wilson theorem revisited
  12. Modular setting
  13. Fractional chromatic number of G F and quantum computing
  14. Concluding remarks and open questions
  15. Proof of Theorem \ref{['theorem: forbidding sunflowers with one but zero']}
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $k \ge 1$ and $m \ge 2$ be integers, and let $L\subseteq [0,k-1]$. Suppose that a finite set system $\mathcal{F}$ of $k$-element sets contains no $L$-sunflower with $m$ petals (in particular, this holds $\mathcal{F}$ it has no $L$-clique of size $m$). Then, writing $a =\min\{i \ge 0: i \notin L\ that avoids intersections of size in $L$. Moreover, this bound is tight for $L$-sunflowers for all

Figures (2)

  • Figure 1: An illustration of $G$ when $k=8$ and $L=\{1,2,3,6\}$ (hence $t=5$). The eight parts of $G$ are depicted by circles and there are edges between two parts if and only if there is a solid line between the two circles.
  • Figure 2: Continuation of the illustration when $k=8$ and $L=\{1,2,3,6\}$. The solid black arrows indicate the number of neighbours in the incoming part of any vertex in the outgoing part. The dashed pink arrows indicate the number of options ($W_{u,v}$) in the incoming part given any two vertices in the two outgoing parts.

Theorems & Definitions (27)

  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5: Füredi's delta-system lemma F78
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8: king2025triply
  • Theorem 1.9
  • Lemma 2.1
  • Theorem 3.1
  • ...and 17 more