  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.
