Let S be a family of 500 different 10-element subsets of the first one hundred positive integers. Prove that it is possible to color each integer 1 through 100 red or blue so that no element of Sis monochromatic. (Note that the elements of S are 10-element subsets of the set (1,2...100).) Bonus question. Prove that it is possible to color the integers from 1 to 1000 red or blue so that no monochromatic arithmetic progression of length 17 is formed. Let S be a family of 500 different 10-element subsets of the first one hundred positive integers. Prove that it is possible to color each integer 1 through 100 red or blue so that no element of Sis monochromatic. (Note that the elements of S are 10-element subsets of the set (1,2...100).) Bonus question. Prove that it is possible to color the integers from 1 to 1000 red or blue so that no monochromatic arithmetic progression of length 17 is formed