## 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 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.

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.

Original language | English |
---|---|

Pages (from-to) | 163-189 |

Number of pages | 27 |

Journal | Combinatorics Probability and Computing |

Volume | 29 |

Issue number | 2 |

DOIs | |

State | Published - 2020 |