Conjecture on minimum \(3\)-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 problem on minimum \(3\)-chromatic hypergraphs (proposed by Erdös and Lovász [1])

Denote by \(n_k^*(r)\) and \(m_k^*(r)\) the minimum number of vertices and edges, respectively, a \((k+1)\)-chromatic \(r\)-graph can have. Determine \(n_k^*(r)\) and \(m_k^*(r)\).

In [1], it was shown that

\(\displaystyle \lim_{r \rightarrow \infty} n_2^*(r)^{1/r} = k \)

\(\displaystyle \lim_{r \rightarrow \infty} m_2^*(r)^{1/r} = k^2. \)

and

\(\displaystyle c_1 \frac{4^r}{r^3} < m_2^*(r) < c_2 r^4 4^r . \)

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.