Mixed pairwise cross intersecting families (I)

An $(n, k_1, \dots, k_t)$-cross intersecting system is a set of non-empty pairwise cross-intersecting families $\mathcal{F}_1\subset{[n]\choose k_1}, \mathcal{F}_2\subset{[n]\choose k_2}, \dots, \mathcal{F}_t\subset{[n]\choose k_t}$ with $t\geq 2$ and $k_1\geq k_2\geq \cdots \geq k_t$. If an $(n, k_1, \dots, k_t)$-cross intersecting system contains at least two families which are cross intersecting freely and at least two families which are cross intersecting but not freely, then we say that the cross intersecting system is of mixed type. All previous studies are on non-mixed type, i.e, under the condition that $n \ge k_1+k_2$. In this paper, we study for the first interesting mixed type, an $(n, k_1, \dots, k_t)$-cross intersecting system with $k_1+k_3\leq n <k_1+k_2$, i.e., families $\mathcal{F}_i\subseteq {[n]\choose k_i}$ and $\mathcal{F}_j\subseteq {[n]\choose k_j}$ are cross intersecting freely if and only if $\{i, j\}=\{1, 2\}$. Let $M(n, k_1, \dots, k_t)$ denote the maximum sum of sizes of families in an $(n, k_1, \dots, k_t)$-cross intersecting system. We determine $M(n, k_1, \dots, k_t)$ and characterize all extremal $(n, k_1, \dots, k_t)$-cross intersecting systems for $k_1+k_3\leq n <k_1+k_2$. We think that the characterization of maximal cross intersecting L-initial families and the unimodality of functions in this paper are interesting in their own, in addition to the extremal result. The most general condition on $n$ is that $n\ge k_1+k_t$. This paper provides foundation work for the solution to the most general condition $n\ge k_1+k_t$.