Splits with forbidden subgraphs

In this paper, we fix a graph Hand ask into how many vertices each vertex of a clique of size n can be "split" such that the resulting graph is H-free. Formally: A graph is an (n, k)-graph if its vertex set is a pairwise disjoint union of n parts of size at most k, such that there is an edge between any two distinct parts. Let f (n, H) = min{k epsilon N : there is an (n, k)-graph G such that H not subset of G}. Barbanera and Ueckerdt [4] observed that f(n, H) = 2 for any graph H that is not bipartite. If a graph His bipartite and has a well-defined Turan exponent, i.e., ex(n, H) = Theta(n(r)) for some r, we show that Omega(n(2/r-1)) = f(n, H) = O(n(2/r-1)log(1/r)n). We extend this result to all bipartite graphs for which upper and a lower Turan exponents do not differ by much. In addition, we prove that f(n, K-2,K- t) = Theta(n(1/3)) for any fixed integer t >= 2. (C) 2021 Elsevier B.V. All rights reserved.