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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
##### Keywords:
van der Waerden numbers; backtracking algorithm
##### Software:
OEIS; tawSolver; UBCSAT
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