Equivalence between Erd\H{o}s-Hajnal and polynomial R\"odl and Nikiforov conjectures

It is well-known that polynomial versions of theorems of R\"odl and Nikiforov, as conjectured by Fox and Sudakov and Nguyen, Scott and Seymour imply the classical Erd\H{o}s-Hajnal conjecture. In this note, we prove that these three conjectures are in fact equivalent, extending several previous particular results in this direction by Fox, Nguyen, Scott and Seymour; Nguyen, Scott and Seymour and Gishboliner and Shapira. We deduce that the family of string graphs satisfies the polynomial R\"odl conjecture. We also derive analogous results for hypergraphs, tournaments, ordered graphs, and colored graphs.