Coloring a complete regular hypergraph

Question

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 $\chi\big((n, [n]^k)\big) = n$?

Answer 1

Notice that a map $c : n \to \kappa$ is a coloring of $(n,[n]^k)$ iff no element of $\kappa$ has $k$ distinct preimages. Hence by the pigeonhole principle $\chi((n,[n]^k)) = \lceil\frac{n}{k-1}\rceil$.

Now for any $k \geq 3$ and $n > 2k-2$ we have $n > 2$, so $\frac{n}{n-1} < 2 \leq k-1$, implying $\frac{n}{k-1} < n-1$.

Thus $\chi((n,[n]^k)) = \lceil\frac{n}{k-1}\rceil < n$.