Quantitative theorems for regular systems of equations
Rado defined a matrix \(L\) with integral coefficients to be partition regular if for every partition of the set of all natural numbers into finitely many classes there exist elements \(x_1,\ldots,x_n\) contained in one class such that \(L(x_1,\ldots,x_n)=0\). [For background cf. \textit{W. A. Deuber}, Developments based on Rado's Dissertation ``Studien zur Kombinatorik'', in: Surveys in combinatorics, Inv. Pap. 12th Br. Comb. Conf., Norwich/UK 1989, Lond. Math. Soc. Lect. Note Ser. 141, 52--74 (1989; Zbl 0768.05001).] It is shown that for \(N\to \infty\) and colorings of \(\{1,\ldots,N\}\) with \(r\) colors a positive fraction of the solutions of the partition regular system \(Lx=0\) in \(\{1,\ldots,N\}\) are monochromatic. The case of partition regular systems for which a density version holds (like arithmetic progressions) is discussed as well as the case of partitions with an arbitrary number of classes.
scientific article; zbMATH DE number 4066893
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