Conjecture on \(3\)-chromatic hypergraphs

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 general conjecture on \(3\)-chromatic hypergraphs (proposed by Erdös and Lovász [1])

Suppose that a hypergraph \(H\) is \(3\)-chromatic but not necessarily uniform. Define \[ f(r)= \min_H \sum_{F \in E{H}} \frac{1}{2^{|F|}} \] where \(H\) ranges over all hypergraphs with minimum edge cardinality \(r\) (i.e., \(\min_F |F| =r\)). Then \[ f(r) \rightarrow \infty \] as \(r \rightarrow \infty\).

Bibliography
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.