A 2-coloring of [1,n] can have (n^2)/22 + O(n) monochromatic Schur   triples, but not less!
  We prove that the minimum number (asymptotically) of monochromatic Schur triples that a 2-coloring of [1,n] can have is (n^2)/22 + O(n). This was solved independently by Tomasz Schoen. 
Aaron Robertson, Doron Zeilberger