We study a combinatorial property of the Parity language (the set of binary words with an even number of 1s). Let \(n\) be a word size, \(\textit{Parity}_n\) be the subset of Parity of words of length \(n\), \(\overline{\textit{Parity}}_n\) be the subset of the complement of Parity of words of length \(n\).

For two subset \(A\) and \(B\), we say that \(B\) has a limit in \(A\) if there is a word \(u\in A\) such that for every position \(i\) there exists a word \(v\) in \(B\) that matches \(u\) on that position (ie. \(u_i = v_i\)).

Let \(p\) be any polynomial. Show: \[\exists A \subseteq \textit{Parity}_n, \ \forall A'\subseteq A, \ |A'|\geq \frac{|A|}{p(n)}, \exists B \subseteq \overline{\textit{Parity}}_n,\] \[\forall B' \subseteq B, \ |B'|\geq \frac{|B|}{p(n)},\ B' \textit{ has a limit in } A'\]

Note that the set \(B\) can be chosen accordingly to the choice of \(A'\).

This property is useful while studying the expressive power of depth-3 boolean circuits in \(\textit{AC}^0\). A similar property has been proved for depth-3 circuits with bounded top fan-in in a recent LICS paper, and can be traced back to Hastad, Jukna and Pudlak. Here, the goal is to extend this combinatorial statement for a bigger class of languages, and for a more general notion of limit to describe precisely the regular languages definable thanks to depth-4 circuits with bounded top fan-in.