\(3\)-chromatic cliques have edges with large intersection
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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.
A problem on \(3\)-chromatic \(r\)-cliques ($100) (proposed by Erdös and Lovász [1], [2])
Let \(H\) be an \(r\)-graph in which every two edges have a nontrivial intersection (that is, \(H\) is an \(r\)-clique). Suppose that \(H\) is \(3\)-chromatic. Is it true that \(H\) contains two edges \(E\) and \(F\) for which \[ |E \cap F| \geq r-2 ? \]Erdösand Lovász [1], [2] proved that there are always two edges \( E\) and \( F\) in such an \( r\)-clique such that
They also showed that in a \( 3\)-chromatic \( r\)-clique there are at least three different values which are the sizes of the pairwise intersection of edges for large enough \( r\).
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1 |
P. Erdös and L. Lovász, Problems and results on 3-chromatic
hypergraphs and some related questions,
Infinite and finite sets (Colloq., Keszthely,
1973; dedicated to P. Erdös on his 60th birthday), Vol. I; Colloq.
Math. Soc. János Bolyai, Vol. 10, 609-627, North-Holland,
Amsterdam, 1975.
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2 | P. Erdös. Problems and results on set systems and hypergraphs. Extremal problems for finite sets (Visegrád, 1991), Bolyai Soc. Math. Stud., 3, pp. 217-227, János Bolyai Math. Soc., Budapest, 1994.
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