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Generalized Ramsey--Turán density for cliques

Jun Gao, Suyun Jiang, Hong Liu, Maya Sankar

Abstract

We study the generalized Ramsey--Turán function $\mathrm{RT}(n,K_s,K_t,o(n))$, which is the maximum possible number of copies of $K_s$ in an $n$-vertex $K_t$-free graph with independence number $o(n)$. The case when $s=2$ was settled by Erd{ő}s, S{ó}s, Bollob{á}s, Hajnal, and Szemerédi in the 1980s. We combinatorially resolve the general case for all $s\ge 3$, showing that the (asymptotic) extremal graphs for this problem have simple (bounded) structures. In particular, it implies that the extremal structures follow a periodic pattern when $t$ is much larger than $s$. Our results disprove a conjecture of Balogh, Liu, and Sharifzadeh and show that a relaxed version does hold.

Generalized Ramsey--Turán density for cliques

Abstract

We study the generalized Ramsey--Turán function , which is the maximum possible number of copies of in an -vertex -free graph with independence number . The case when was settled by Erd{ő}s, S{ó}s, Bollob{á}s, Hajnal, and Szemerédi in the 1980s. We combinatorially resolve the general case for all , showing that the (asymptotic) extremal graphs for this problem have simple (bounded) structures. In particular, it implies that the extremal structures follow a periodic pattern when is much larger than . Our results disprove a conjecture of Balogh, Liu, and Sharifzadeh and show that a relaxed version does hold.
Paper Structure (17 sections, 19 theorems, 93 equations, 2 figures, 1 table)

This paper contains 17 sections, 19 theorems, 93 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Organization.
  3. Notation.
  4. Reduction to Weighted Graphs
  5. Upper bound
  6. Lower bound
  7. Understanding the extremal weighted graphs
  8. Proof of \ref{['sup1']}
  9. Proof of \ref{['sup2']}
  10. Proof of \ref{['sup3']} and \ref{['sup4']} for $a=2$
  11. Proof of \ref{['sup3']} and \ref{['sup4']} for all $a$
  12. Proof of \ref{['sup5']}
  13. Eventual periodicity and counterexamples to Conjecture \ref{['conj']}
  14. Eventual Periodicity
  15. The Remaining Positive Results
  16. ...and 2 more sections

Key Result

Theorem 1.3

Conjecture conj is false when $2s\leq t\leq 2.08s$ for sufficiently large $s$. Given $t-2\geq s\geq 3$, Conjecture conj is true if $t>s^2(s-1)/2+s$, $t=s+2$, or $s=3,4$.

Figures (2)

  • Figure 4.1: The optimizations in the proof of $s=4$. Red edges have weight $1/2$ and black edges have weight 1.
  • Figure 4.2: Counterexamples to \ref{['conj']} for $s=5$ and $t\in \{10,11\}$. Red edges have weight $1/2$ and black edges have weight 1.

Theorems & Definitions (44)

  • Definition 1.1
  • Conjecture 1.2: balogh2017on
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 34 more