Every n-vertex graph of VC-dimension ≤ d has a homogeneous set of size at least n^{(C d)^{-d}} for an absolute constant C.
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A Single-Exponential Erd\H{o}s--Hajnal Bound for Graphs of Bounded VC-Dimension
Every n-vertex graph of VC-dimension ≤ d has a homogeneous set of size at least n^{(C d)^{-d}} for an absolute constant C.