Section 18.1 Introduction
Ramsey theory, a branch of combinatorics, explores conditions under which order must appear within seemingly random or chaotic structures. Named after British mathematician Frank P. Ramsey, the theory addresses various problems in mathematics that involve finding regular patterns within disordered systems. Its principles have far–reaching implications across disciplines such as computer science, logic, and even the natural sciences, making it a fundamental area of study in understanding complexity and order in mathematics.
Among the early results in Ramsey theory is van der Waerden's theorem, which states that for any positive integers \(r,k\) there exists a number \(N\) such that for any \(r\)–colouring of the integers \(\{1,2,\ldots,N\}\text{,}\) it contains an arithmetic progression of length \(k\) where all the terms are of the same colour. [18.6.1] The smallest such \(N\) is called the van der Waerden number \(W(r,k)\text{.}\)
Similarly, for any positive integers \(r, k_0, k_1, \ldots, k_{r-1}\text{,}\)the van der Waerden number \(w = W(r; k_0, k_1, \ldots, k_{r-1})\) is the minimum integer such that any \(r\)–colouring of the set \(\{1,2,\ldots,w\}\) contains an arithmetic progression of length \(k_i\) of colour \(i\) for some \(i\text{.}\) [18.6.2]
The primary objective of this paper is to present a newly computed van der Waerden number, which marks an advancement in our knowledge of these numbers. This computation contributes to the ongoing effort to explore the range of van der Waerden numbers.