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 $n\ge k_1+k_t$

rr-cross tt-intersecting families via necessary intersection points

Given integers $r\geq 2$ and $n,t\geq 1$ we call families $\mathcal{F}_1,\dots,\mathcal{F}_r\subseteq\mathscr{P}([n])$ $r$-cross $t$-intersecting if for all $F_i\in\mathcal{F}_i$, $i\in[r]$, we have $\vert\bigcap_{i\in[r]}F_i\vert\geq t$. We obtain a strong generalisation of the classic Hilton-Milner theorem on cross intersecting families. In particular, we determine the maximum of $\sum_{j\in [r]}\vert\mathcal{F}_j\vert$ for $r$-cross $t$-intersecting families in the cases when these are $k$-uniform families or arbitrary subfamilies of $\mathscr{P}([n])$. Only some special cases of these results had been proved before. We obtain the aforementioned theorems as instances of a more general result that considers measures of $r$-cross $t$-intersecting families. This also provides the maximum of $\sum_{j\in [r]}\vert\mathcal{F}_j\vert$ for families of possibly mixed uniformities $k_1,\ldots,k_r$.Comment: 13 page

On the number of maximal intersecting k-uniform families and further applications of Tuza's set pair method

We study the function $M(n,k)$ which denotes the number of maximal $k$-uniform intersecting families $F\subseteq \binom{[n]}{k}$. Improving a bound of Balogh at al. on $M(n,k)$, we determine the order of magnitude of $\log M(n,k)$ by proving that for any fixed $k$, $M(n,k) =n^{\Theta(\binom{2k}{k})}$ holds. Our proof is based on Tuza's set pair approach. The main idea is to bound the size of the largest possible point set of a cross-intersecting system. We also introduce and investigate some related functions and parameters.Comment: 11 page

Multiple cross-intersecting families of signed sets

A k-signed r-set on[n] = {1, ..., n} is an ordered pair (A, f), where A is an r-subset of [n] and f is a function from A to [k]. Families A1, ..., Ap are said to be cross-intersecting if any set in any family Ai intersects any set in any other family Aj. Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of [n]. Our aim is to generalise Hilton's bound to one for families of k-signed r-sets on [n]. The main tool developed is an extension of Katona's cyclic permutation argument.peer-reviewe