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Random embeddings of bounded-degree trees with optimal spread
- Part of
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- Journal:
- Combinatorics, Probability and Computing , First View
- Published online by Cambridge University Press:
- 13 October 2025, pp. 1-12
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- Article
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A seminal result of Komlós, Sárközy, and Szemerédi 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 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.
