Linear hereditary quasirandomness

Conlon, Fox and Sudakov (Math. Proc. Cambridge Philos. Soc. 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 \(m≥1\) edges, and fix \(p∈(0,1)\). If every subset \(S\) of an \(n\)-vertex graph \(G\) contains \(p^m|S|^r±δn^r\) labelled injective copies of \(H\), then \( e_G(A,B)-p|A||B|≤ CH,pδn^2\) for all \(A,B⊆ V(G)\), 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 \(2e(G)/n^2=p+O(n⁻²)\).