Canonical partition relations for \((m,p,c)\)-systems (Q1356665)

In order to develop an infinite version of the coloring theorems on finite sums in [\textit{W. Deuber}, Partitionen und lineare Gleichungssysteme, Math. Z. 133, 109-123 (1973; Zbl 0254.05011)] and [\textit{H. Lefmann}, A canonical version for partition regular systems of linear equations, J. Comb. Theory, Ser. A 41, 95-104 (1986; Zbl 0583.05006)], a number of results about sequences of \((m,p,c)\)-sets are presented. Here, an \((m,p,c)\)-system on \((x_1,\dots,x_m)\) is a set of positive integers \[ \Biggl\{\sum^m_{i= t+1}\lambda_ix_i+ cx_t:t\in\{1,\dots, m\} \& \lambda_i\in\{0,\dots, p\}\Biggr\}. \] Letting \(\overline x\in V=\prod^\infty_{n=1}\mathbb{N}^n\), we can imagine \(S(\overline x,n)\) as being the \((n,n\cdot n!,n!)\)-set of \(\overline x(n)\), and \[ \text{FS}(\langle S(\overline x,n)\rangle_{n\in M})= \Biggl\{\sum_{n\in F}w_n:F\subseteq M \& F\text{ finite }\& \forall n(w_n\in S(\overline x,n))\Biggr\}. \] Each sequence \(\overline x\in V\) generates its own sequence of \((m,p,c)\)-sets and hence its own sequence of finite sums from elements of \((m,p,c)\)-sets. The first few sections present some results about comparing these sets of finite sums from one sequence \(\overline x\) to that of a `refinement' \(\overline y\). Using these results, and assuming a conjecture of [\textit{P. Erdös} and \textit{R. Graham}, Old and new problems and results in combinatorial number theory, Monographies No. 28 de l'Enseignment Mathematique (1980; Zbl 0434.10001)], it is shown that all colorings of the finite sums of some sufficiently nice \(\overline x\) are well-behaved on the finite sums of some refinement \(\overline y\) of \(\overline x\).