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

Let Cₖ denote a cycle on k vertices. Pascadi (2026) established a removal lemma for C₂ₖ₊₁ in graphs with few C₂ₖ-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.