Let A1,A2,... be a sequence of infinite sets such that ∣Ai∩Aj∣≤2 for i \not\equal{}j. Show that the sequence of indices can be divided into two disjoint sequences i1<i2<... and j1<j2<... in such a way that, for some sets E and F, |A_{i_n} \cap E|\equal{}1 and |A_{j_n} \cap F|\equal{}1 for n\equal{}1,2,... .
P. Erdos combinatorics proposedcombinatorics