Resolve Scott’s chromatic-number conjecture for graphs with chromatic number at least three

Prove or disprove Scott’s conjecture that every n-vertex graph without isolated vertices and chromatic number χ contains an odd induced subgraph of order at least n/(2χ) for graphs with χ≥3.

Background

Scott’s conjecture proposes a chromatic-number-dependent lower bound for the order of a largest odd induced subgraph. The paper notes that Wang and Wu disproved the conjecture for bipartite graphs, corresponding to chromatic number two, while confirming it for all line graphs.

The conjecture remains unresolved for graphs with chromatic number at least three, so determining its validity in that range is an explicit open problem.