Claims

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2013_08_21_dudek_mubayi: Dudek and Mubayi apply Shearer's independence bound beside a vertex neighborhood and get f(n) >= c sqrt(n log n / log log n) for Problem 620; refereed in J. Graph Theory 76 (2014).

2024_01_04_mubayi_verstraete: Theorem 1 of Mubayi and Verstraete, f_s(n) = O(sqrt(n) log n) for each fixed s at least 3, bounds f(n) of Problem 620 from above; refereed in Bull. Lond. Math. Soc. 57 (2025).

2026_07_17_morris_sahasrabudhe_verstraete: An arXiv preprint of 17 July 2026 claims that the Erdős–Rogers function f_{s,s+1}(n) has order sqrt(n log n) for every s at least 2, which at s = 3 determines the order of f(n) up to constants; unrefereed, so claimed.

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