An approximate version of the Loebl-Komlos-Sos conjecture

TL;DR

This paper proves an approximate version of the Loebl-Komlos-Sos conjecture for large graphs, establishing an asymptotic bound on the Ramsey number of trees, which advances understanding of tree embeddings in dense graphs.

Contribution

It provides an asymptotic proof of the conjecture for large graphs and derives a bound on the Ramsey number of trees, extending previous combinatorial results.

Findings

Proves an approximate version of the Loebl-Komlos-Sos conjecture for large n.

Establishes that the Ramsey number r(T_k,T_m) is at most k+m+o(k+m).

Demonstrates the conjecture's implications for asymptotic bounds in graph theory.

Abstract

Loebl, Komlos, and Sos conjectured that if at least half of the vertices of a graph G have degree at least some natural number k, then every tree with at most k edges is a subgraph of G. Our main result is an approximate version of this conjecture for large enough n=|V(G)|, assumed that n=O(k). Our result implies an asymptotic bound for the Ramsey number of trees. We prove that r(T_k,T_m)\leq k+m+o(k+m),as k+m tends to infinity.