Minimum degree and diversity in intersecting antichains (Q2036583)

For a family \( \mathcal{F} \subset 2^{[n]}\) and \(i \in [n]\) define \(\mathcal{F}(i)=\{F\backslash \{i\}: i\in F\in \mathcal{F}\}\). The minimum degree of \(\mathcal{F}\) is defined as \(\min_{i\in [n]}\vert \mathcal{F}(i)\vert \). Let \(n,l\) be positive integers such that \(n>2l+1>1\), \(X\) be an \(n\)-element set and \(\mathcal{F}\) an antichain \(\mathcal{F}\subset 2^{X}\). Kiselev, Kupavskii and Patkós conjectured that if \(\vert F\cup G\vert \leq 2l+1\) for all \(F,G\in \mathcal{F}\) then the minimum degree of \(\mathcal{F}\) is no more than \(\binom{n-1}{l-1}\), the minimum degree of \(\binom{[n]}{l}\). In this paper it is shown that the conjecture holds for \(n\geq l^{3}+l^{2}+3l/2\) or \(l=1\) or 2. The analogous problem for the case \(\vert F\cap G\vert \geq t\) was also solved. The last section contains two conjectures of some independent interest.