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

Abstract

Given integers 𝑟 ⩾ 2 and 𝑛, 𝑡 ⩾ 1 we call families  1 , … ,We obtain a strong generalisation of the classic Hilton-Milner theorem on cross-intersecting families. In particular, we determine the maximum of ∑ 𝑗∈[𝑟] | 𝑗 | for 𝑟-cross 𝑡-intersecting families in the cases when these are 𝑘-uniform families or arbitrary subfamilies of 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 ∑ 𝑗∈[𝑟] | 𝑗 | for families of possibly mixed uniformities 𝑘 1 , … , 𝑘 𝑟 .