Abstract. Schur’s theorem states that in any [Formula: see text]-coloring of the set of integers [Formula: see text] there is a monochromatic solution to [Formula: see text], provided [Formula: see text] is sufficiently large. Abbott and Wang studied the size of the largest subset of [Formula: see text] such that there is a [Formula: see text]-coloring avoiding a monochromatic [Formula: see text]. This led to the exploration of related problems, such as the minimum number of monochromatic [Formula: see text] in [Formula: see text]-colorings of [Formula: see text] and the probability threshold for a random subset of [Formula: see text] to have a monochromatic [Formula: see text] in any [Formula: see text]-coloring. In this paper, we study natural generalizations of these problems to products [Formula: see text], in deterministic, random, and randomly perturbed environments.

Abstract. We address a problem which is a generalization of Turán-type problems recently introduced by Imolay, Karl, Nagy, and Váli. Let [Formula: see text] be a fixed graph and let [Formula: see text] be the union of [Formula: see text] edge-disjoint copies of [Formula: see text], namely [Formula: see text], where each [Formula: see text] is isomorphic to a fixed graph [Formula: see text] and [Formula: see text] for all [Formula: see text]. We call a subgraph [Formula: see text] multicolored if [Formula: see text] and [Formula: see text] share at most one edge for all [Formula: see text]. Define [Formula: see text] to be the maximum value [Formula: see text] such that there exists [Formula: see text] on [Formula: see text] vertices without a multicolored copy of [Formula: see text]. We show that [Formula: see text] and that all extremal graphs are close to a blow-up of the 5-cycle. This bound is tight up to the linear error term.