2021/01/08 by Frankl, P., Katona, G. O. H.
#05D05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2101.03015
Let n > k > t ≥ j ≥ 1 be integers. Let X be an n-element set, X\choose k the collection of its k-subsets. A family \mathcal F ⊂ X\choose k is called t-intersecting if |F ∩ F'| ≥ t for all F, F' ∈ \mathcal F. The j'th shadow ∂j \mathcal F is the collection of all (k - j)-subsets that are contained in some member of~\mathcal F. Estimating |∂j \mathcal F| as a function of |\mathcal F| is a widely used tool in extremal set theory. A classical result of the second author (Theorem \refth:1.3) provides such a bound for t-intersecting families. It is best possible for |\mathcal F| = 2k - t\choose k. Our main result is Theorem \refth:1.4 which gives an asymptotically optimal bound on |∂j \mathcal F| / |\mathcal F| for |\mathcal F| slightly larger, e.g., |\mathcal F| > \frac32 2k - t\choose k. We provide further improvements for |\mathcal F| very large as well.