A new lower bound for van der Waerden numbers

Abstract In this paper we prove a new recurrence relation on the van der Waerden numbers, w ( r , k ) . In particular, if p is a prime and p ≤ k then w ( r , k ) > p ⋅ w r − r p , k − 1 . This recurrence gives the lower bound w ( r , p + 1 ) > p r − 1 2 p when r ≤ p , which generalizes Berlekamp’s theorem on 2-colorings, and gives the best known bound for a large interval of r . The recurrence can also be used to construct explicit valid colorings, and it improves known lower bounds on small van der Waerden numbers.

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