We call a finite, simple, undirected graph $G=(V,E)$ an $n$-*Erdos graph* if there are $n$ subsets $S\_1,\ldots, S\_n$ of $V$ such that


1. $V = \bigcup\_{n=1}^n S\_n$;
2. each $S\_k$ has $n$ elements for $k\in\{1,\ldots, n\}$;
3. $\{a,b\}\in E$ implies that there is $k\in\{1,\ldots,n\}$ such that $a, b\in S\_k$;
4. $k\neq j\in \{1,\ldots, n\}$ imply $|S\_k\cap S\_j|\leq 1$.


One version of the [Erdos-Faber-Lovasz conjecture](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture) says that if $G$ is an $n$-Erdos graph, then $\chi(G) \leq n$.


We say that $G$ is a *weak* $n$-*Erdos graph* if items 1.-3. above are satisfied, but not necessarily number 4. 


Is there $k\in \mathbb{R}$ with $k>1$ such that for all $n\in \mathbb{N}$ we have that if $G$ is a weak $n$-Erdos graph, then $\chi(G) \leq k\cdot n$? If yes, what is the smallest such $k$?


(Note: for $n=3$ it is possible to find a weak $n$-Erdos graph $G$ with $\chi(G) = 4$, maybe I add the example later on. So if such a $k$ exists we have $k\geq 4/3$.)



Weak Erdos graphs
A complete graph with $\Theta(n^{3/2})$ vertices can be a weak $n$-Erdos graph. 


Let $m$ be the largest integer satisfying $m(m-1)/2 \le n$. Consider $m$ disjoint copies of sets $V\_1, \dots, V\_m$ each with $\lfloor n/2 \rfloor$ elements. Let the sets $S\_i$s be
$$\{S\_1, S\_2, \dots, S\_{m(m-1)/2}\} = \{V\_1 \cup V\_2, V\_1 \cup V\_3, \dots, V\_{m-1} \cup V\_m\}.$$
Then the complete graph on the set of vertices $V\_1 \cup \dots \cup V\_m$ is a weak $n$-Erdos graph with $mn$ vertices. Clearly, the chromatic number of this graph is $mn \sim cn^{3/2}$. 


