A combinatorial property of Fréchet iterated filters

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