For integers k ≥3 and r ≥ 2, let W(k, r) be the smallest positive integer n such that for any coloring of any block of n consecutive integers with r colors, the block of integers contains a monochromatic k-term arithmetic-progression. In 1927, van der Waerden [3] showed that W(k, r) exists for any k ≥ 3 and r ≥ 2. Determining W(k, r), however, is still a difficult challenge.
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机译：对于整数K≥3和r≥2，设有w（k，r）是最小的正整数n，使得对于任何带有r颜色的N个连续整数的任何块的任何颜色，整数块包含单色k术语算术 - 进出。 1927年，van der Waerden [3]显示W（K，R）存在任何K≥3和r≥2.然而，确定W（K，R）仍然是一个艰难的挑战。
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