About maximal number of edges in hypergraph-clique with chromatic number 3

Let $H = (V,E)$ be a hypergraph. By the chromatic number of a hypergraph $H = (V,E)$ we mean the minimum number $\chi(H)$ of colors needed to paint all the vertices in $V$ so that any edge $e \in E$ contains at least two vertices of some different colors. Finally, a hypergraph is said to form a clique, if its edges are pairwise intersecting. In 1973 Erd\H{o}s and Lov\'asz noticed that if an $n$-uniform hypergraph $H = (V,E)$ forms a clique, then $\chi(H) \in \{2,3\}$. They untoduced following quantity. $$M(n) = \max \{|E|: \exists {\rm an} n-{\rm uniform} {\rm clique} H = (V,E) {\rm with} \chi(H) = 3\}.$$ Obviously such definition has no sense in the case of $\chi(H) = 2$. Theorem 1 (P. Erdos, L. Lovasz} The inequalities hold $$n!(e-1) \le M(n) \le n^n.$$ Almost nothing better has been done during the last 35 years. At the same time, another quantity $r(n)$ was introduced by Lovasz r(n) = \max \{|E|: ~ \exists {\rm an} ~ n-{\rm uniform} ~ {\rm clique} ~ H = (V,E) ~ {\rm s.t.} ~ \tau(H) = n\}, where $ \tau(H) $ is the {\it covering number} of $ H $, i.e., \tau(H) = \min \{|f|: ~ f \subset V, ~ \forall ~ e \in E ~ f \cap e \neq \emptyset\}. Clearly, for any $n$-uniform clique $ H $, we have $ \tau(H) \le n $, and if $ \chi(H) = 3 $, then $ \tau(H) = n $. Thus, $ M(n) \le r(n) $. Lov\'asz noticed that for $ r(n) $ the same estimates as in Theorem 1 apply and conjectured that the lower estimate is best possible. In 1996 P. Frankl, K. Ota, and N. Tokushige disproved this conjecture and showed that $ r(n) \ge (\frac{n}{2})^{n-1} $. We discovered a new upper bound for the r(n) (so for M(n) too). Theorem 2. M(n) \leq r(n) \le c n^{n-1/2} \ln n. $$, where c is a constant.