Given D and γ > 0, whenever c > 0 is sufficiently small and n sufficiently large, if G is a family of D-degenerate graphs of individual orders at most n, maximum degrees at most cn log n , and total number of edges at most (1 − γ) n 2 , then G packs into the complete graph Kn. Our proof proceeds by analysing a natural random greedy packing algorithm.

Abstract. We prove that for any pair of constants ε > 0 and ∆ and for n sufficiently large, every family of trees of orders at most n, maximum degrees at most ∆, and with at most n 2 edges in total packs into K (1+ε)n . This implies asymptotic versions of the Tree Packing Conjecture of Gyárfás from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof.

Loebl, Komlós, and Sós conjectured that if at least half of the vertices of a graph G have degree at least some k ∈ N, 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 , Tm) ≤ k + m + o(k + m), as k + m → ∞.

Abstract. We find, for all sufficiently large n and each k, the maximum number of edges in an n-vertex graph which does not contain k + 1 vertex-disjoint triangles.This extends a result of Moon [Canad. J. Math. 20 (1968), 96-102] which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the Corrádi-Hajnal Theorem.