Edge-statistics on large graphs

TL;DR

This paper investigates the maximum number of induced subgraphs with fixed size and edge count in large graphs, proposing a conjecture and providing partial bounds supported by probabilistic and combinatorial methods.

Contribution

It introduces the Edge-statistics conjecture for induced subgraph counts and proves bounds that support its validity, advancing understanding of subgraph distributions in large graphs.

Findings

The density of subgraphs with fixed size and edges is bounded away from 1 by a constant.

For most pairs (k, l), only a polynomially small fraction of k-sets have exactly l edges.

An upper bound of (1/2 + o_k(1)) * binomial(n, k) is established for l=1.

Abstract

The inducibility of a graph $H$ measures the maximum number of induced copies of $H$ a large graph $G$ can have. Generalizing this notion, we study how many induced subgraphs of fixed order $k$ and size $\ell$ a large graph $G$ on $n$ vertices can have. Clearly, this number is $\binom{n}{k}$ for every $n$, $k$ and $\ell \in \left \{0, \binom{k}{2} \right\}$. We conjecture that for every $n$, $k$ and $0 < \ell < \binom{k}{2}$ this number is at most $\left(1/e + o_k(1) \right) \binom{n}{k}$. If true, this would be tight for $\ell \in \{1, k-1\}$. In support of our `Edge-statistics conjecture' we prove that the corresponding density is bounded away from $1$ by an absolute constant. Furthermore, for various ranges of the values of $\ell$ we establish stronger bounds. In particular, we prove that for `almost all' pairs $(k, \ell)$ only a polynomially small fraction of the $k$-subsets of $V(G)$ has exactly $\ell$ edges, and prove an upper bound of $(1/2 + o_k(1))\binom{n}{k}$ for $\ell = 1$. Our proof methods involve probabilistic tools, such as anti-concentration results relying on fourth moment estimates and Brun's sieve, as well as graph-theoretic and combinatorial arguments such as Zykov's symmetrization, Sperner's theorem and various counting techniques.