Leírás
As one of the most fundamental theorems in extremal set theory, Erdős-Ko-Rado Theorem
determined the upper bound of size of an intersecting k-family. This theorem has various generalizations,
such as the Hilton–Milner theorem, the Ray-Chaudhuri–Wilson theorem, the r-wise intersection theorem,
and the Complete Intersection theorem. The above-mentioned theorems all consider the maximum size
of an intersecting family with some additional conditions. In this talk, instead of focusing on the size of the
intersecting family, we consider the sum of size of all intersections of an intersecting family F, denoted by w(F).
Although the values of w(F) and |F| are not directly correlated, in the following theorem, by using cyclic permutation,
we can still prove that when F is a star, w(F) reaches its maximum value.
Further, we generalize this result for crossing-intersecting families.