A new spectral argument shows that very dense graphs far from unions of cliques have least eigenvalue at most -n^{1/4-epsilon}, and this yields max-cut surpluses of order n^{1.01} and m^{0.5001} for H-free graphs.
Quadratic forms on graphs
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An Alon-Boppana--type bound for very dense graphs, with applications to max-cut
A new spectral argument shows that very dense graphs far from unions of cliques have least eigenvalue at most -n^{1/4-epsilon}, and this yields max-cut surpluses of order n^{1.01} and m^{0.5001} for H-free graphs.