Strong Ramsey theorems for Steiner systems

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Abstract
It is shown that the class of partial Steiner ( k , l ) (k,\,l) -systems has the edge Ramsey property, i.e., we prove that for every partial Steiner ( k , l ) (k,\,l) -system G \mathcal {G} there exists a partial Steiner ( k , l ) (k,\,l) -system H \mathcal {H} such that for every partition of the edges of H \mathcal {H} into two classes one can find an induced monochromatic copy of G \mathcal {G} . As an application we get that the class of all graphs without cycles of lengths 3 3 and 4 4 has the edge Ramsey property. This solves a longstanding problem in the area.

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