I came across the following pigeonhole principle exercise.
Suppose that the numbers $1,\ldots, 10$ are arranged around a circle in an arbitrary order. Show there must be three consecutive entries whose sum is at least $18$.
There is a very nice short proof of this, and also, there is an arrangement such that the sum of any three consecutive is at most $18$.
We can define $f(n,k)$ as the minimum, over all arrangements of $\{1,\ldots, n\}$, of the maximum sum of any $k$ consecutive entries. So the exercise above gives $f(10,3)=18$. I thought briefly about some simple things, like $k=1, 2, n-1, n$
I was wondering if anyone knows anything about this problem in general. We could think of it as finding a vertex labeling of a hypergraph where the maximum sum of vertex labels in an edge is minimized. The hypergraph in consideration is a $k$-uniform tight cycle on $n$ vertices. We could of course make it more general and consider other hypergraphs.
Is there a name for such a labeling or for this optimization problem?
Let $G$ be a $k$-uniform hypergraph on $n$ vertices and let $\mathcal{S}$ be the set of all bijections from $V(G)$ to $[n]$. For $f\in\mathcal{S}$ and $e \in E(G)$, let $f(e) = \sum_{v\in e} f(v)$. The problem is to find $$\min_{f\in \mathcal{S}}\max_{e\in E(G)}f(e)$$