Geodesic
Tower Gaps in Multicolour Ramsey Numbers
Forum of Mathematics, Sigma, Tome 11 (2023)

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Resolving a problem of Conlon, Fox and Rödl, we construct a family of hypergraphs with arbitrarily large tower height separation between their $2$-colour and q-colour Ramsey numbers. The main lemma underlying this construction is a new variant of the Erdős–Hajnal stepping-up lemma for a generalized Ramsey number $r_k(t;q,p)$, which we define as the smallest integer n such that every q-colouring of the k-sets on n vertices contains a set of t vertices spanning fewer than p colours. Our results provide the first tower-type lower bounds on these numbers. 
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     author = {Quentin Dubroff and Ant\'onio Gir\~ao and Eoin Hurley and Corrine Yap},
     title = {Tower {Gaps} in {Multicolour} {Ramsey} {Numbers}},
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Quentin Dubroff; António Girão; Eoin Hurley; Corrine Yap. Tower Gaps in Multicolour Ramsey Numbers. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.89

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