Monochromatic sumsets

LEMMA . At most (kn)'g"ulk k-sets Sc [ n ] have IP(S)I <u . Proof. Let a,< . . . < a k denote the elements of S. Call i doubling if P(a, , . . ., a i) has double the size of P(a, , . . ., a i 1 ) . There are at most lg u doubling i. Hence there are at most k'g" choices for doubling positions i and at most n'g" choices for the values a i . If i is not doubling then a i = x y, where x, y E P(a, , . . ., a i , ) P(S) so there are at most u 2 choices for a i . Proof of Theorem . Two-color [n] randomly. The expected number of k-sets S with P(S) monochromatic is then Note