Counting subgraphs in locally dense graphs

TL;DR

This paper investigates the number of subgraphs in locally dense graphs, proving the KNRS conjecture for new classes of graphs, exploring stability conditions, and connecting these problems to semidefinite optimization and copositive matrices.

Contribution

The paper proves the KNRS conjecture for new graph classes, establishes stability results linking quasirandomness to subgraph counts, and introduces a weakened conjecture related to degree-regularity.

Findings

Proved the conjecture for graphs preserved under natural gluing operations.

Established that approximate equality in the conjecture implies quasirandomness.

Extended the conjecture to nearly degree-regular graphs and proved it for more graph families.

Abstract

A graph $G$ is said to be $p$-locally dense if every induced subgraph of $G$ with linearly many vertices has edge density at least $p$. A famous conjecture of Kohayakawa, Nagle, R\"odl, and Schacht predicts that locally dense graphs have, asymptotically, at least as many copies of any fixed graph $H$ as are found in a random graph of edge density $p$. In this paper, we prove several results around the KNRS conjecture. First, we prove that certain natural gluing operations on $H$ preserve this property, thus proving the conjecture for many graphs $H$ for which it was previously unknown. Secondly, we study a stability version of this conjecture, and prove that for many graphs $H$, approximate equality is attained in the KNRS conjecture if and only if the host graph $G$ is quasirandom. Finally, we introduce a weakening of the KNRS conjecture, which requires the host graph to be nearly degree-regular, and prove this conjecture for a larger family of graphs. Our techniques reveal a surprising connection between these questions, semidefinite optimization, and the study of copositive matrices.