Set systems with k-wise \(\mathcal {L} \)-intersections and \( \mathcal {L} \)-differencing Sperner families modulo prime powers

Abstract

   In this paper, we obtain the following improved upper bound on the size of k-wise \( \mathcal {L} \)-intersecting families and \( \mathcal {L} \)-differencing Sperner families modulo prime powers by employing linear algebra methods:

1. (1)

   Let \( k \ge 2 \) be an integer, p be a prime, \( q = p^{ \alpha } \) be a prime power with \( \alpha \ge 1 \), and let \( \mathcal {L} \subseteq \{0, 1, \dots , q-1\} \) be a subset with size \( s > 0 \). Suppose that \( \mathcal {F} \) is a family of subsets of [n] such that \( \left| F \right| \notin \mathcal {L} \pmod {q} \) for each \( F \in \mathcal {F} \) and \( \left| F_{i_1} \cap F_{i_2} \cap \cdots \cap F_{i_k} \right| \in \mathcal {L} \pmod {q} \) for every collection of k distinct subsets in \( \mathcal {F} \). Then

   $$ \left| \mathcal {F} \right| \le \left( k-1 \right) \left[ {{n-1} \atopwithdelims (){q-1}}+ {{n-1} \atopwithdelims (){q-2}}+ \cdots + {{n-1} \atopwithdelims (){0}} \right] . $$

   If in addition there exists an integer \( t < p \) such that \( \left| F \right| \in \left\{ q-t, q-t+1, \dots q-1 \right\} \pmod {q} \) for each \( F \in \mathcal {F} \), then

   $$ \left| \mathcal {F} \right| \le \left( k-1 \right) \left[ {{n-1} \atopwithdelims (){q-1}}+ {{n-1} \atopwithdelims (){q-2}}+ \cdots + {{n-1} \atopwithdelims (){p - t -1}} \right] . $$

   This result not only gives an improvement to a theorem by G. Hegedüs and a theorem by Z. Xu and C. H. Yip, but also extends a theorem by L. Babai and P. Frankl.

2. (2)

   Let p be a prime, \( q = p^{ \alpha } \) be a prime power with \( \alpha \ge 1 \), and \( \mathcal {L} = \{l_1, l_2, \dots , l_s\} \) be a subset of \( \{1, 2, \dots , q-1\} \). Suppose that \( \mathcal {F} \) is a family of subsets of [n] satisfying that \( \left| F \setminus F' \right| \in \mathcal {L} \pmod {q} \) for all distinct pairs \( F, F' \in \mathcal {F} \). Then

   $$ \left| \mathcal {F} \right| \le \left( {\begin{array}{c}n-1\\ q-1\end{array}}\right) + \left( {\begin{array}{c}n-1\\ q-2\end{array}}\right) +\cdots +\left( {\begin{array}{c}n-1\\ 0\end{array}}\right) . $$

   This result extends a theorem by Z. Xu and C. H. Yip to the case of general \( \mathcal {L} \) with size \( \left| \mathcal {L} \right| = s \).