Sharp thresholds for hypergraph regressive Ramsey numbers

The f-regressive Ramsey number Rreg f (d, n) is the minimum N such that every colouring of the d-tuples of an N-element set mapping each x1, . . . , xd to a colour ≤ f(x1) contains a min-homogeneous set of size n, where a set is called min-homogeneous if every two d-tuples from this set that have the same smallest element get the same colour. If f is the identity, then we are dealing with the standard regressive Ramsey numbers as defined by Kanamori and McAloon. In this paper we classifiy the growth-rate of the regressive Ramsey numbers for hypergraphs in dependence of the growth-rate of the parameter function f. The growth-rate has to be measured against the scale of fast-growing Hardy functions Fα indexed by towers of exponentiation in base ω. Our results give a sharp classification of the thresholds at which the f-regressive Ramsey numbers undergoe a drastical change in growth-rate. The case of graphs has been treated of Lee, Kojman, Omri and Weiermann. We extend their result...

• Type: Journal
• Year: 2011
• Where: JCT
• Authors: Lorenzo Carlucci, Gyesik Lee, Andreas Weiermann
• Keywords: JCT 2011, Ramsey Numbers, Regressive Ramsey, Regressive Ramsey Numbers