what is hyper-tournament?

Question

I know the definition of the tournament ( a directed graph obtained after assigning direction to edges of the complete graph). I tried on search on internet did not get anything.

Question: What is hyper-graphs means?

Motivation : This term is mentioned in this paper

Answer 1

Before the definition of a hypertournament, here are some preliminaries. A hypergraph is a generalization of a graph in the following sense:

• A graph is defined by a set $V$ of vertices and a set $E$ of edges which are unordered pairs $\{v,w\}$ with $v, w \in V$.
• A hypergraph is defined by a set $V$ of vertices and a set $E$ of "hyperedges" (often just edges) which are subsets of $V$ of arbitrary size.
• Often (as in this paper) we talk about $k$-uniform hypergraphs: in this case, all hyperedges must have size $k$. That is, each edge is a set $\{v_1, v_2, \dots, v_k\}$ where $v_1, v_2, \dots, v_k$ are distinct elements of $V$.

In particular, an ordinary graph is a $2$-uniform hypergraph.

Just as a complete graph is a graph which includes all $\binom{|V|}{2}$ possible edges, a complete $k$-uniform hypergraph is defined to be the hypergraph whose edges are all $\binom{|V|}{k}$ possible subsets of $V$ of size $k$.

Finally, a $k$-hypertournament (as defined in the paper referenced in the question) is the corresponding generalization of a tournament: it is obtained from a complete $k$-uniform hypergraph by assigning directions to each hyperedge.

What does it mean to assign a direction to a hyperedge? It means that we replace the set $\{v_1, v_2, \dots, v_k\}$ by an ordered $k$-tuple of these vertices, in some order: $(v_{\pi(1)}, v_{\pi(2)}, \dots, v_{\pi(k)}\}$ for some permutation $\pi$ of $\{1,2,\dots,k\}$.

For example:

• All ordinary tournaments are $2$-hypertournaments: a directed edge from $v$ to $w$ is just an ordered pair $(v,w)$.
• One possible $3$-hypertournament on the vertices $\{1,2,3,4\}$ is the one whose oriented hyperedges are the triples $\{(3,2,1), (2,1,4), (3,1,4), (3,4,2)\}$. This is one out of $(3!)^4$ different possibilities.
• If you held a Hearts competition between $n$ players in which each of the $\binom n4$ sets of players play a game together, you could record the $4$-tuples $(1^{\text{st}}\text{ place}, 2^{\text{nd}}\text{ place}, 3^{\text{rd}}\text{ place}, 4^{\text{th}}\text{ place})$ for each game played, and that would be a $4$-hypertournament. (Assuming there are no ties.)