Deviation probabilities for arithmetic progressions and other regular discrete structures

Abstract

Let the random variable X ∶= e( [B]) count the number of edges of a hypergraph  induced by a random m element subset B of its vertex set. Focussing on the case that  satisfies some regularity condition we prove bounds on the probability that X is far from its mean. It is possible to apply these results to discrete structures such as the set of k-term arithmetic progressions in the cyclic group Z N . Furthermore, we show that our main theorem is essentially best possible and we deduce results for the case B ∼ B p is generated by including each vertex independently with probability p.