For any set $X$ and positive integer $k$ denote by $[X]^k$ the set of subsets $S\subseteq X$ such that $|S|=k$.

Let $H=(V,E)$ be a hypergraph such that for every $e\in E$ we have $|e|\geq 2$. A map $c:V\to \kappa$, where $\kappa$ is a cardinal, is said to be a *(hypergraph) coloring* if for all $e\in E$ the restriction $c|_e$ is not constant. By $\chi(H)$ we denote the smallest cardinal $\kappa$ such that there is a coloring $c:V\to \kappa$.

Given any positive integer $n$, we consider it as a finite cardinal $n = \{0,\ldots,n-1\}$. It is easy to see that if $a, b$ are positive integers with $a < b \leq 2a$ then $\chi\big((2a, [2a]^b)\big) = 2$: color the even members of $2a$ with $0$, and the odd members with $1$.

Given a fixed positive integer $k>2$, is there always $n>2k-2$ such that $\chi\big((n, [n]^k)\big) = n$?