On the parametric limit superior of a sequence of analytic sets.

Abstract

Let $A_x$ stand for $x$-section of a set $A\subset2^\omega\times2^\omega$. We prove that any sequence $A_j\subset2^\omega\times2^\omega$, $j\in\omega$ of analytic sets, with uncountable $\limsup_{j\in H}A_x^j$ for all $x\in2^\omega$ and $H\in [\omega]^\omega$ admits a perfect set $P\subset2^\omega$ and $H\subset [\omega]^\omega$ with uncountable $\bigcap_{j\in H}A_x^j$ for all $x\in P$. This is a parametric version of the Komjath theorem [2].