2020/05/14 by Liu, Xizhi, Mubayi, Dhruv
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2005.06999
A fundamental result in extremal set theory is Katona's shadow intersection theorem, which extends the Kruskal-Katona theorem by giving a lower bound on the size of the shadow of an intersecting family of k-sets in terms of its size. We improve this classical result and a related result of Ahlswede, Aydinian, and Khachatrian by proving tight bounds for families that can be quite small. For example, when k=3 our result is sharp for all families with n points and at least 3n-7 triples. Katona's theorem was extended by Frankl to families with matching number s. We improve Frankl's result by giving tight bounds for large n.