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  "body": "\n## Definition\n\nA hypergraph is a pair $H = (V, E)$ where $V$ is a [[set|0SJ3RX9TXA]] and $E$ is a [[family|05Q6G4EH2G]] of nonempty subsets of $V$.\nThe elements of $V$ are the vertices, and the members of $E$ are the hyperedges.\nLet $k$ be a positive integer.\nThe hypergraph $H$ is $k$-uniform if each hyperedge has exactly $k$ elements.\n\n## Remarks\n\nSome authors allow repeated hyperedges or the empty hyperedge; others require $E$ to be a set.\nA [[graph|0JD5C82QMB]] is the same thing as a $2$-uniform hypergraph without repeated edges.\nExample: the [[Fano plane|QQ8GW1ASZM]], with the $7$ points as vertices and the $7$ lines as hyperedges, is a $3$-uniform hypergraph, since each line has $3$ points.\nNon-example: $V = \\set{1, 2, 3}$ with $E = \\set{\\set{1, 2}, \\set{1, 2, 3}}$ is a hypergraph that is not $k$-uniform for any $k$, because its hyperedges have sizes $2$ and $3$.\nEvery hypergraph has an incidence matrix, with a row for each vertex and a column for each hyperedge.\n",
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