The Size-Ramsey Number of Trees

If G and H are graphs, let us write G → (H)2 if G contains a monochromatic copy ofH in any 2-colouring of the edges of G. The size-Ramsey number re(H) of a graph H is the smallest possible number of edges a graph G may have if G→ (H)2. Suppose T is a tree of order |T | ≥ 2, and let t0, t1 be the cardinalities of the vertex classes of T as a bipartite graph, and let ∆(T ) be the maximal degree of T . Moreover, let ∆0, ∆1 be the maxima of the degrees of the vertices in the respective vertex classes, and let β(T ) = t0∆0+t1∆1. Beck [7] proved that β(T )/4 ≤ re(T ) = O{β(T )(log |T |)}, improving on a previous result of his [6] stating that re(T ) ≤ ∆(T )|T |(log |T |). In [6], Beck conjectures that re(T ) = O{∆(T )|T |}, and in [7] he puts forward the stronger conjecture that re(T ) = O{β(T )}. Here, we prove the first of these conjectures, and come quite close to proving the second by showing that re(T ) = O{β(T ) log∆(T )}. 1991 Mathematics Subject Classification 05C 55, 05C 05, 05C 80 ∗Current address: Department of Combinatorics and Optimisation, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 The first author was partially supported by NSERC. The second author was partially supported by FAPESP (Proc. 93/0603-1) and by CNPq (Proc. 300334/93–1).