Number of sizes of edge intersections in a \(3\)-chromatic \(r\)-graph

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 the edge-intersections of \(3\)-chromatic hypergraphs (proposed by Erdös and Lovász [1])

Let \(g(r)\) denote the least integer such that in any \(3\)-chromatic \(r\)-graph \(H\), the cardinalities \(|E\cap F|\), for edges \(E, F\) in \(H\), take on at least \(g(r)\) values.
Is it true that \[ g(r) \rightarrow \infty \] as \(r \rightarrow \infty\)?
Is it true that \[ g(r) = r-2? \]


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.