On the size of shadow-added intersecting families

Abstract

Let n≥2k−1>1, [n]={1,2,…,n}. For a family F of k-subsets of [n] let ∂F be the immediate shadow (cf. Definition 1.1) of F. Suppose that |F∩F′|≥2 for all F,F′∈F. We conjecture that |F|+|∂F|≤3n−2k−2+n−2k−3 and prove it for n=2k−1, n≥3(k−1) and also for k≤10. This problem is somewhat unusual but we exhibit deep connections to the Erdős–Ko–Rado Theorem and to the Erdős Matching Conjecture. Some related problems are also considered.