Random embeddings of bounded degree trees with optimal spread

TL;DR

This paper presents a new regularity-free method for embedding bounded degree trees into graphs with high minimum degree, improving on previous approaches that relied on complex regularity lemmas.

Contribution

It introduces an alternative construction that avoids the regularity lemma by leveraging a black-box version of the Komlós-Sárközy-Szemerédi theorem, simplifying the embedding process.

Findings

Provides a regularity-free embedding technique for bounded degree trees.

Shows almost all small subgraphs inherit the minimum degree condition.

Achieves optimal spread distribution for embeddings.

Abstract

A seminal result of Koml\'os, S\'ark\"ozy, and Szemer\'edi states that any n-vertex graph G with minimum degree at least (1/2 + {\alpha})n contains every n-vertex tree T of bounded degree. Recently, Pham, Sah, Sawhney, and Simkin extended this result to show that such graphs G in fact support an optimally spread distribution on copies of a given T, which implies, using the recent breakthroughs on the Kahn-Kalai conjecture, the robustness result that T is a subgraph of sparse random subgraphs of G as well. Pham, Sah, Sawhney, and Simkin construct their optimally spread distribution by following closely the original proof of the Koml\'os-S\'ark\"ozy-Szemer\'edi theorem which uses the blow-up lemma and the Szemer\'edi regularity lemma. We give an alternative, regularity-free construction that instead uses the Koml\'os-S\'ark\"ozy-Szemer\'edi theorem (which has a regularity-free proof due to Kathapurkar and Montgomery) as a black-box. Our proof is based on the simple and general insight that, if G has linear minimum degree, almost all constant sized subgraphs of G inherit the same minimum degree condition that G has.