Splits with forbidden subgraphs

In this paper, we fix a graph H and ask into how many vertices each vertex of clique size n can be “split” such that the resulting is H-free. Formally: A an (n,k)-graph if its set pairwise disjoint union parts at most k, there edge between any two distinct parts. Letf(n,H)=min⁡{k∈N:there G H⊈G}. Barbanera Ueckerdt [4] observed f(n,H)=2 for not bipartite. If bipartite has well-defined Turán exponent, i.e., ex(n,H)=Θ(nr) some r, show Ω(n2/r−1)=f(n,H)=O(n2/r−1log1/r⁡n). We extend result to all graphs which upper lower exponents do differ by much. addition, prove f(n,K2,t)=Θ(n1/3) fixed integer t≥2.