In this post, it was mentioned that a long time ago, Ajtai, Kolmós, Simonovits, and Szemerédi announced a proof that for sufficiently large $k$, every $k$-vertex tree $T$ is a subgraph of every graph $G$ with average degree $>(k-2)$. However, this was never published.
I was curious if there is a high-level sketch of how their argument should go. I have heard that general outlines were discussed at conferences, but if there is anything in writing I'd appreciate seeing it.
Looking at Section 1.2 of this paper, it is suggested that their argument involved "an extension of the regularity lemma", is there a more precise description of the kind of regularity lemma they sought?
