Combinatorics in one-based and related structures

We consider some extremal combinatorial questions for bipartite graphs definable in stable one-based (and related) structures. We show that they satisfy both strong Erdős-Hajnal property and linear Zarankiewicz. We also show that the same is true for both collapsed and uncollapsed Hrushovski’s ``ab initio’’ constructions, and discuss some connections to Zilber’s trichotomy principle. For strong Erdős-Hajnal, we show that in fact it holds in a more general class of $1$-semi-equational theories.