The sparse regularity method for graphs with few even-cycle homomorphisms

Abstract

Let $C_k$ denote a cycle on $k$ vertices. Pascadi (2026) established a removal lemma for $C_{2k+1}$ in graphs with few $C_{2k}$-homomorphisms under a dense-pairs condition, using Schatten--von Neumann norms and entropy methods. We give a short proof of a version with more restrictive partition and diagonal requirements, extending the combinatorial approach of Conlon, Fox, Sudakov and Zhao (2021) without using Pascadi's Schatten-norm regularity framework. We illustrate the method with a known sparse removal statement for odd product equations in finite groups.