Partitions and sums of integers with repetition

AbstractA partition of N is called “admissible” provided some cell has arbitrarily long arithmetic progressions of even integers in a fixed increment. The principal result is that the statement “Whenever {Ai}i < r is an admissible partition of N, there are some i < r and some sequence 〈xn〉n < ω of distinct members of N such that xn + xm ϵ Ai whenever {m, n} ⊆ ω″ is true when r = 2 and false when r ⩾ 3