... for every H ∈B(m, ∆). In [1] and [2] the problemof finding minimum M = M(m) for which there exists an (m, ∆)-universal graph with Medges is investigated. Here we apply Theorems 1.3 and 1.4 to ... tower type for Szemer´edi’s uniformity lemma, GAFA,Geom. Funct. Anal. 7 (1997), 322-337.[5] R. L. Graham, V. R¨odl and A. Ruci´nski, On bipartite graphs with linear Ramseynumbers, Combinatorica, ... blowing up G(k, t) was to obtain graphs with more vertices than n1andstill having -holes in large subgraphs. Next we consider a random “contraction” of G(k, t) to obtain graphs with fewer than...