A skew version of the Loebl-Komlós-Sós conjecture

Loebl, Komlós, and Sós conjectured that any graph such that at least half of its vertices have degree at least k contains every tree of order at most k + 1. We propose a skew version of this conjecture. We consider the class of trees of order at most k + 1 of given skew, that is, such that the sizes of the colour classes of the trees have a given ratio. We show that our conjecture is asymptotically correct for dense graphs. The proof relies on the regularity method. Our result implies bounds on Ramsey number of several trees of given skew.