Combinatorial proofs of Gowers' FIN$_k$ and FIN$_k^\pm$ theorem
  A combinatorial proof of a pigeonhole principle of Gowers is found along with its symmetric and approximate version, FIN$_k^\pm$ theorem. The proofs do not use of the concept of ultrafilter. 
Ryszard Frankiewicz, S{\l}awomir Szczepaniak
math.CO