The Erdős-Hajnal conjecture on high-girth high-chromatic subgraphs holds throughout every fixed polynomial edge-density regime relative to chromatic number, with an explicit double-exponential bound on the required chromatic number.
Walczak, Triangle-free geometric intersection graphs with no large independent sets, Discrete Comput
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The Erd\H{o}s-Hajnal High-Girth Subgraph Conjecture Holds in the Polynomial Chromatic-Sparsity Regime
The Erdős-Hajnal conjecture on high-girth high-chromatic subgraphs holds throughout every fixed polynomial edge-density regime relative to chromatic number, with an explicit double-exponential bound on the required chromatic number.