Pseudo-arithmetic sets and Ramsey theory (Q1826855)

For \(n\in{\mathbb N}\), \([n]\) denotes the set \(\{1,2,\dots,n\}\); \([n]^k\) denotes the collection of subsets of \([n]\) of cardinality \(k\) (not the set of ordered \(k\)-tuples). A set \(A\) of natural numbers is pseudo-arithmetic if all pairwise differences of its elements are divisible by the minimum positive difference between members, or if its cardinality is less than 3; \(A\) is a pseudo-arithmetic super set if all of its subsets are pseudo-arithmetic. Theorem 4.1. Let \(c,k\) be positive integers. For any \(c\)-colouring \(f:[{\mathbb N}]^k\to[c]\) of \([{\mathbb N}]^k\) there exists a monochromatic, infinite, pseudo-arithmetic super set \(A\subseteq{\mathbb N}\), i.e., \(f\) is constant on \([A]^k\). Theorem 4.2. Let \(c,k,n\) be positive integers. There is a natural number \(P(n;c,k)\) such that, for any \(c\)-colouring \(f:[P(n;c,k)]^k\to c\), there is a monochromatic, pseudo-arithmetic set \(A\) with \(| A| =n\). It is shown that \(P(3;2,2)=13\). The Schur number \(S(r)\) is the smallest natural number \(n\) such that any \(r\)-colouring of \([n]\) has a monochromatic Schur triple \(\{x,y,x+y\}\) [cf., \textit{I. Schur}, Über die Kongruenz \(x^m+y^m\equiv z^m \pmod p\). Deutsche Math. Ver. 25, 114--117 (1916; JFM 46.0193.02)], where the author permits equality of \(x\) and \(y\); \(S_d(r)\) is the smallest natural number \(n\) such that any \(r\)-colouring of \([n]\) has a divisible monochromatic Schur triple \(\{x,y,x+y\}\), i.e., wherein \(x| y\) or \(y| x\). Theorem 6.1. For every positive integer \(r\) there exists an integer \(n_0\) such that, for \(n\geq n_0\), there exist, in any \(r\)-colouring of \([n]\), \(x,y,z\in[n]\) bearing the same colour, and such that \(x+y=z\) and \(x| y\). An example is provided of a 3-colouring of \([612]\) with no divisible Schur triples, whose existence implies that \(S_d(3)\geq613\) and \(P(3;3,2)\geq614\); the existence of an example implying that \(S_d(3)\geq864\) is announced; verification of these properties would appear to require computer assistance.