Hedgehog (hypergraph)

In the mathematical theory of hypergraphs, a hedgehog is a 3-uniform hypergraph defined from an integer parameter $$t$$. It has $$t+\tbinom{t}{2}$$ vertices, $$t$$ of which can be labeled by the integers from $$1$$ to $$t$$ and the remaining $$\tbinom{t}{2}$$ of which can be labeled by unordered pairs of these integers. For each pair of integers $$i,j$$ in this range, it has a hyperedge whose vertices have the labels $$i$$, $$j$$, and $$\{i,j\}$$. Equivalently it can be formed from a complete graph by adding a new vertex to each edge of the complete graph, extending it to an order-3 hyperedge.

The properties of this hypergraph make it of interest in Ramsey theory.