On Non-Empty Cross-Intersecting Families

r$r$‐Cross t$t$‐intersecting families via necessary intersection points

“…Now it is natural to ask for the maximum of $$\sum _{i\in [r]}\vert \mathcal {F}_i\vert$$ taken over all non‐empty $$r$$‐cross $$t$$‐intersecting families $$\mathcal {F}_1,\ldots ,\mathcal {F}_r$$. In this regime there are several partial results concerning the maximum sum of sizes of $$r$$‐cross $$t$$‐intersecting families for specific instances of $$r$$ and $$t$$, starting with theorems by Hilton [25] and by Hilton and Milner [26] and continued, for instance, in [6, 20, 21, 24, 31, 32, 34, 35] (also see the references therein). We determine $$\sum _{i\in [r]}\vert \mathcal {F}_i\vert$$ for every $$r \geqslant 2$$ and $$t \geqslant 1$$ for both uniform families and non‐uniform families (see Corollary 1.3 and Corollary 1.5), generalising a result by Frankl and Wong H.W.…”

“…For $$r=2$$ and $$t\geqslant 1$$ Corollary 1.3 was proved by Frankl and Kupavskii [20]. For $$t=1$$ and $$r \geqslant 2$$ Corollary 1.3 was shown very recently in independent work by Shi, Frankl and Qian [31], where they deduce it from a result about two families by an elegant application of the Kruskal–Katona theorem [27, 28].…”

“…Sometimes, a weaker definition of ‘cross intersecting’ is used in the literature: for $$r,t,n\in \mathbb {N}$$, we say that families $$\mathcal {F}_1,\dots ,\mathcal {F}_r\subseteq \mathcal {P}([n])$$ are pairwise $$r$$ ‐cross $$t$$ ‐intersecting if for all $$i,j\in [r]$$ and all $$F_i\in \mathcal {F}_i,F_j\in \mathcal {F}_j$$, we have $$\vert F_i\cap F_j\vert \geqslant t$$. In [31], Shi, Frankl and Qian posed three problems regarding the maximum sum of sizes of pairwise $$r$$‐cross $$t$$‐intersecting families. Theorem 1.2 solves all of these problems for $$r$$‐cross $$t$$‐intersecting families (and slightly larger $$n$$).…”