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We prove a general large sieve statement in the context of random walks onsubgraphs of a given graph. This can be seen as a generalization of previouslyknown results where one performs a random walk on a group enjoying a strongspectral gap property. In such a context the point is to exhibit a stronguniform expansion property for a suitable family of Cayley graphs on quotients.In our combinatorial approach, this is replaced by a result of Alon--Roichmanabout expanding properties of random Cayley graphs. Applying the generalsetting we show e.g., that with high probability (in a strong explicit sense)random coloured subsets of integers contain monochromatic (non-empty) subsetssumming to zero, or that a random coloring of the edges of a complete graphcontains a monochromatic triangle.

Expanders graphs and sieving in combinatorial structures