Maximum edges in a \(3\)-chromatic \(r\)-clique
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A hypergraph \( H\) consists of a vertex set \( V\) together with a family \( E\) of subsets of \( V\), which are called the edges of \( H\). A \( r\)-uniform hypergraph, or \( r\)-graph, for short, is a hypergraph whose edge set consists of \( r\)-subsets of \( V\). A graph is just a special case of an \( r\)-graph with \( r=2\).
A hypergraph \( H\) is said to be \( k\)-chromatic if the vertices of \( H\) can be colored in \( k\) colors so that every edge has at least \( 2\) colors.
An \( r\)-graph is said to be a clique if every two edges have a nontrivial intersection. An unexpected fact about \( 3\)-chromatic \( r\)-graphs is that there are only finitely many \( 3\)-chromatic \( r\)-cliques.
A problem on maximum \(3\)-chromatic hypergraphs (proposed by Erdös and Lovász [1])
Determine the maximum number \(M(r)\) of edges a \(3\)-chromatic \(r\)-clique can have.Erdös and Lovász [1] proved that