Forbidden induced subgraphs for bounded pp-intersection number

A graph GG has pp-intersection number at most dd if it is possible to assign to every vertex uu of GG, a subset S(u)S(u) of some ground set UU with |U|=d|U|=d in such a way that distinct vertices uu and vv of GG are adjacent in GG if and only if |S(u)∩S(v)|≥p|S(u)∩S(v)|≥p. We show that every minimal forbidden induced subgraph for the hereditary class G(d,p)G(d,p) of graphs whose pp-intersection number is at most dd, has order at most (2d+1)2(2d+1)2, and that the exponential dependence on dd in this upper bound is necessary. For p∈{d−1,d−2}p∈{d−1,d−2}, we provide more explicit results characterizing the graphs in G(d,p)G(d,p) without isolated/universal vertices using forbidden induced subgraphs.