Abstract
AbstractFor graphs and a family of graphs , we write to denote that every blue‐red coloring of the edges of contains either a blue copy of , or a red copy of each . For integers and , let denote the family of all trees with edges and maximum degree at most . We prove that for each , there exist constants such that if and , then with high probability. This is a random version of a well‐known result of Chvátal from 1977. The proof combines a stability argument with the embedding of trees in expander graphs. Furthermore, the proof of the stability result is based on a sparse random analogue of the Erdős–Sós conjecture for trees with linear size and bounded maximum degree, which may be of independent interest.
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