A combinatorial property of Fréchet iterated filters

For all m ⩾ 1 let N m be the filter on ω m obtained from the Fréchet filter on ω by iterating the Fubini product operation. One easily checks that N m is a Σ 2 m 0 subset of P ( ω m ) and the main result of this work states that in any Π 2 m 0 subset of P ( ω m ) containing N m one can find m + 1 elements with empty intersection.