Intersections and distinct intersections in cross-intersecting families

Let F,G be two cross-intersecting families of k-subsets {1,2,…,n}. F∧G, I(F,G) denote the all intersections F∩G with F∈F,G∈G, and distinct F≠G,F∈F,G∈G, respectively. For a fixed T⊂{1,2,…,n}, let ST family {1,2,…,n} containing T. In present paper, we show that |F∧G| is maximized when F=G=S{1} for n≥2k2+8k, while surprisingly |I(F,G)| F=S{1,2}∪S{3,4}∪S{1,4,5}∪S{2,3,6} G=S{1,3}∪S{2,4}∪S{1,4,6}∪S{2,3,5} n≥100k2. The maximum number in t-intersecting determined n≥3(t+2)3k2 as well.