Long gaps in sieved sets

For each prime p, let Ip⊂Z/pZ denote a collection of residue classes modulo p such that the cardinalities |Ip| are bounded and about 1 on average. We show that for sufficiently large x, the sifted set {n∈Z:n(modp)∉Ipforallp≤x} contains gaps of size x(logx)δ depends only on the densitiy of primes for which Ip≠∅. This improves on the "trivial'' bound of ≫x. As a consequence, for any non-constant polynomial f:Z→Z with positive leading coefficient, the set {n≤X:f(n)composite} contains an interval of consecutive integers of length ≥(logX)(loglogX)δ for sufficiently large X, where δ>0 depends only on the degree of f.