Preprint 2026-04
Akbar Davoodi, Diana Piguet, Hanka Rada, Nicolás Sanhueza-Matamala:
The asymptotic version of the Erdős–Sós conjecture and beyond
Abstract:
Klimošová, Piguet, and Rozhoň conjectured that any graph with minimum degree k/2 and sufficiently many vertices of degree k should contain all trees with k edges. We prove an asymptotic version of this conjecture for dense host graphs. We obtain interesting corollaries: the first is an asymptotic version of the Erdős--Sós conjecture for dense host graphs, which works without any bounded-degree restriction on the guest trees. Secondly, by leveraging recent results by Pokrovsky, we can translate our results to sparse host graphs in the case of bounded-degree guest trees.
