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Find, with proof, the minimum positive integer n with the following property: for any coloring of the integers {1, 2, . . . , n} using the colors red and blue (that is, assigning the color “red” or “blue” to each integer in the set), there exist distinct integers a, b, c between 1 and n, inclusive, all of the same color, such that 2a + b = c.

 Sep 12, 2021
 #1
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The minimum vaue of of n is 10.

 Sep 12, 2021
 #2
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This is not correct. If the red group is (1,2,3) and the blue group is (4,5,6,7,8,9,10), then the desired condition is not possible.

broww  Sep 12, 2021

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