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Tight bounds towards a conjecture of Gallai

We prove that for n > k ≥ 3, if G is an n -vertex graph with chromatic number k but any of its proper subgraphs has smaller chromatic number, then G contains at most n − k + 3 copies of a clique of size k − 1. This answers a problem of Abbott and Zhou and provides a tight bound on a conjecture of Gallai.