Gupta, PranshuPranshuGuptaMogge, YannickYannickMoggePiga, SimónSimónPigaSchülke, BjarneBjarneSchülke2023-03-312023-03-312023-02-20Bulletin of the London Mathematical Society 55(3): 1447-1458 (2023)http://hdl.handle.net/11420/15075Given integers (Formula presented.) and (Formula presented.) we call families (Formula presented.) (Formula presented.) -cross (Formula presented.) -intersecting if for all (Formula presented.), (Formula presented.), we have (Formula presented.). We obtain a strong generalisation of the classic Hilton–Milner theorem on cross-intersecting families. In particular, we determine the maximum of (Formula presented.) for (Formula presented.) -cross (Formula presented.) -intersecting families in the cases when these are (Formula presented.) -uniform families or arbitrary subfamilies of (Formula presented.). 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 (Formula presented.) -cross (Formula presented.) -intersecting families. This also provides the maximum of (Formula presented.) for families of possibly mixed uniformities (Formula presented.).en1469-21202023314471458Wileyhttps://creativecommons.org/licenses/by-nc-nd/4.0/MathematicsJournal Article10.1112/blms.12803Journal Article