Edge-statistics on large graphs

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 a large
graph G on n vertices can have. Clearly, this number is n
k

for every n, k and ∈ {0, k
2

}. We conjecture
that for every n, k and 0 << k
2

this number is at most (1/e + ok(1))n
k

. If true, this would be tight for
∈ {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 we establish
stronger bounds. In particular, we prove that for ‘almost all’ pairs (k, ) only a polynomially small fraction
of the k-subsets of V(G) have exactly edges, and prove an upper bound of (1/2 + ok(1))n
k

for = 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.