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Supersaturation for subgraph counts

Jonathan Cutler, JD Nir, A. J. Radcliffe

Abstract

The classic extremal problem is that of computing the maximum number of edges in an $F$-free graph. In the case where $F=K_{r+1}$, the extremal number was determined by Turán. Later results, known as supersaturation theorems, proved that in a graph containing more edges than the extremal number, there must also be many copies of $K_{r+1}$. Alon and Shikhelman introduced a broader class of problems asking for the maximum number of copies of a graph $T$ in an $F$-free graph. In this paper, we determine some of these generalized extremal numbers and prove supersaturation results for them.

Supersaturation for subgraph counts

Abstract

The classic extremal problem is that of computing the maximum number of edges in an -free graph. In the case where , the extremal number was determined by Turán. Later results, known as supersaturation theorems, proved that in a graph containing more edges than the extremal number, there must also be many copies of . Alon and Shikhelman introduced a broader class of problems asking for the maximum number of copies of a graph in an -free graph. In this paper, we determine some of these generalized extremal numbers and prove supersaturation results for them.

Paper Structure

This paper contains 7 sections, 29 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Cliques without cliques
  3. Cliques without stars
  4. Stars without stars
  5. Stars without cliques
  6. Supersaturation for cliques without stars
  7. Many stars, no $K_{r+1}$

Key Result

Theorem 1.1

Let $r\geq 1$ be an integer and let $\varepsilon>0$. Then there exists $n_0=n_0(r,\varepsilon)$ such that if $G$ is a graph on $n\geq n_0$ vertices and then $G$ contains $K_{r+1}(b)$ for some $b\geq \varepsilon\log n/(2^{r+1}(r-1)!)$.

Figures (1)

  • Figure 1: Graphs of $f(\rho)$ and $g(\rho)$ with $t=6$

Theorems & Definitions (48)

  • Theorem 1.1: Erdős-Stone
  • Theorem 1.2: Bollobás
  • Theorem 1.3
  • Theorem 1.4: Moon-Moser
  • Theorem 1.5: Nikiforov
  • Theorem 1.6
  • proof
  • Theorem 1.7: Chase
  • Theorem 1.8: Wood, Engbers-Galvin
  • proof
  • ...and 38 more