On rainbow arithmetic progressions
The authors study a variant of the celebrated theorem of van der Waerden on arithmetic progressions: coloring the set \(\{1,2,\dots , n\}\) by \(r\) colors, can we find an arithmetic progression with length \(k\) so that all its elements are colored in distinct colors? In the main part of the paper the authors investigate the case \(k=r=3\). They prove a conjecture of \textit{V. Jungić} [Exp. Math. 9, No. 3, 467--471 (2000; Zbl 0965.05093)], showing if each color class has cardinality at least \({n+4\over 6}\) then there exists a three-term arithmetic progression with distinct colors. Furthermore for \(k\geq 5\) they construct a coloring of \(\{1,2,\dots , n\}\), \(n=k(2m)\), for which each color class is \(n/k\) and there is no \(k\)-term arithmetic progression with distinct colors. In the case \(k=4\) it is constructed a coloring of \(\{1,2,\dots , n\}\), \(n=10m+1\), with the smallest color class of size \((n-1)/5\) and there is no 4-term arithmetic progression with distinct colors.
scientific article; zbMATH DE number 2038347