Random embeddings of bounded-degree trees with optimal spread

A seminal result of Komlós, Sárközy, and Szemerédi states that any n -vertex graph G with minimum degree at least (1/2+α )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 Komlós-Sárközy-Szemerédi theorem which uses the blow-up lemma and the Szemerédi regularity lemma. We give an alternative, regularity-free construction that instead uses the Komlós-Sárközy-Szemerédi 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.