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Some more van der Waerden numbers. (English) Zbl 1317.11020
Summary: The van der Waerden number \(w(k; t_{0}, t_{1}, \dots , t_{k-1})\) is the smallest positive integer \(n\) such that every \(k\)-coloring of the sequence \(1, 2,\ldots, n\) yields a monochromatic arithmetic progression of length \(t_{i}\) for some color \(i \in \{0, 1, \ldots , k-1\}\). In this paper, we propose a problem-specific backtracking algorithm for computing van der Waerden numbers \(w(k; t_{0}, t_{1}, \ldots , t_{k-1})\) with \(t_{0} = t_{1} = \ldots = t_{j-1} = 2\), where \(k \geq j+2\), and \(t_{i} \geq 3\) for \(i \geq j\). We report some previously unknown van der Waerden numbers using this method. We also report the exact value of the previously unknown van der Waerden number \(w(2; 5, 7)\).

MSC:
11B25 Arithmetic progressions
05D10 Ramsey theory
Software:
OEIS; tawSolver; UBCSAT
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