Real life motivation. In my younger son's class, there are $18$ students. His teacher provided $18$ time slots for the parents of each child to have a 30-minute conversation of their kid's progress in school and asked the parents to provide $4$ possible time slots that would be convenient for them. I wondered how probable it is that given all the choices, the teacher can select a convenient time slot for the parents of every child so that everyone can have the "progress conversation."
Formal version. We regard every $n\in\omega = \mathbb{N}$ as an ordinal,say that is $0 = \emptyset$ and $n = \{0, \ldots, n-1\}$ for $n > 0$. Foran $n, k \in\mathbb{N}$ we let$n$-tuple $[n]^k$ be the collection$(S_1,\ldots,S_n)$ of $k$-element subsets of $n$ (so that $[n]^k = \emptyset$ whenever $k > n$). We say that ${\cal S}\subseteq [n]^k$$\{1,\ldots,n\}$ has the satisfiability property if $|{\cal S}|=n$ and there is a bijectionpermutation $\varphi:{\cal S}\to n$ such that$\sigma \in \mathfrak{S}_n$ with $\sigma(i)\in S_i$ for all $T\in{\cal S}$ we have $$\varphi(T) \in T.$$ We denote the selection of $n$-element collections of $[n]^k$ by $\big[[n]^k\big]^n$$i=1,\ldots,n$.
Given an integer $n>1$, what is the smallest $k$ (in terms of $n$) such that at least half of the elements$n$-tuples $(S_1,\ldots,S_n)$ of $\big[[n]^k\big]^n$$k$-element subsets of $\{1,\ldots,n\}$ have the satisfiability property? If no exact formula can be given, is there an approximation for when $n\to\infty$?
