Linear hereditary quasirandomness

Abstract

Conlon, Fox and Sudakov (\emph{Math. Proc. Cambridge Philos. Soc.} \textbf{164} (2018), 385--399) conjectured that hereditary subgraph counts force quasirandomness with linear dependence on the counting error. We prove this for every graph containing an edge. Let \(H\) have \(r\) vertices and \(m\ge1\) edges, and fix \(p\in(0,1)\). If every subset \(S\) of an \(n\)-vertex graph \(G\) contains \(p^m|S|^r\pmδn^r\) labelled injective copies of \(H\), then \(\lvert e_G(A,B)-p|A||B|\rvert\le C_{H,p}δn^2\) for all \(A,B\subseteq V(G)\), where \(e_G(A,B)\) counts ordered adjacent pairs. The proof extracts averaged rooted counts and represents each nonlinear term by a linear map in one selected edge factor. Composing two such maps gives an edge substitution with a superlinear cut-norm bound. A separate star expansion gives stronger local estimates for triangle-free graphs. The linear order is best possible even when \(2e(G)/n^2=p+O(n^{-2})\).