Spread blow-up lemma with an application to perturbed random graphs

TL;DR

This paper introduces a spread version of the blow-up lemma, enabling probabilistic control over spanning subgraphs in super-regular graphs, and applies it to analyze Hamilton cycle powers in perturbed random graphs.

Contribution

It develops a spread blow-up lemma that combines combinatorial and probabilistic techniques, extending the classical blow-up lemma with new applications.

Findings

Established a spread measure over copies of spanning graphs in super-regular pairs.

Proved an approximate threshold for powers of Hamilton cycles in perturbed random graphs.

Connected combinatorial lemmas with probabilistic methods to solve conjectures.

Abstract

Combining ideas of Pham, Sah, Sawhney, and Simkin on spread perfect matchings in super-regular bipartite graphs with an algorithmic blow-up lemma, we prove a spread version of the blow-up lemma. Intuitively, this means that there exists a probability measure over copies of a desired spanning graph $H$ in a given system of super-regular pairs which does not heavily pin down any subset of vertices. This allows one to complement the use of the blow-up lemma with the recently resolved Kahn-Kalai conjecture. As an application, we prove an approximate version of a conjecture of B\"ottcher, Parczyk, Sgueglia, and Skokan on the threshold for appearance of powers of Hamilton cycles in perturbed random graphs.