On the Density of a Graph and its Blowup

TL;DR

This paper investigates the minimal density of certain blowup subgraphs in graphs with fixed triangle density, providing bounds and insights that relate to quasi-random graph properties and open conjectures.

Contribution

It proves that for graphs with a given triangle density, there exists a blowup size where the subgraph density matches that of a random graph, advancing understanding of graph quasi-randomness.

Findings

Existence of a blowup size t with density close to that in G(n,p)

Different behavior observed in skewed blowups

Raises conjectures and discusses applications

Abstract

The theorem of Chung, Graham, and Wilson on quasi-random graphs asserts that of all graphs with edge density p, the random graph G(n,p) contains the smallest density of copies of K_{t,t}, the complete bipartite graph of size 2t. Since K_{t,t} is a t-blowup of an edge, the following intriguing open question arises: Is it true that of all graphs with triangle density p^3, the random graph G(n,p) contains the smallest density of K_{t,t,t}, which is the t-blowup of a triangle? Our main result gives an indication that the answer to the above question is positive by showing that for some blowup, the answer must be positive. More formally we prove that if G has triangle density p^3, then there is some 2 <= t <= T(p) for which the density of K_{t,t,t} in G is at least p^{(3+o(1))t^2}, which (up to the o(1) term) equals the density of K_{t,t,t} in G(n,p). We also consider the analogous question on skewed blowups, showing that somewhat surprisingly, the behavior there is different. We also raise several conjectures related to these problems and discuss some applications to other areas.