Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). If $\kappa$ is a cardinal, we say that a map $c:E\to \kappa$ is an *edge coloring* if whenever $e\_1,e\_2\in E$ with $e\_1\cap e\_2\neq \emptyset$ then $c(e\_1)\neq c(e\_2)$. The smallest cardinal $\kappa$ such that there is an edge coloring $c:E\to \kappa$ is called the *edge chromatic number* of $H$, denoted by $\chi\_e(H)$. 


We say that $H=(V,E)$ is a *dense linear hypergraph* if 


1. $V \notin E$,
2. $\bigcup E = V$,
3. whenever $e\_1\neq e\_2 \in E$ then $|e\_1\cap e\_2| \leq 1$, and
4. given $a\neq b\in V$ there is $e\in E$ with $\{a,b\}\in e$.


Given a positive integer $k$, is there a dense linear hypergraph $H= (V,E)$ with $V$ finite and $\chi\_e(H) < 1/k\cdot |V|$?



Edge coloring in dense linear hypergraphs
Take a finite projective plane $π=\{P,L\}$ of order $n$, and remove all the points from a line $l$. Let the removed points be $p\_1,p\_2,...p\_{n+1}$. The hypergraph $H$ defined by such an incidence structure has $n^2$ vertices, and every edge has $n$ elements.


Property 1 and 2 are obvious.


Property 3 follows from the fact that projective planes are linear hypergraphs. 


Property 4 follows from the definition of a projective plane. 


A coloring with $n+1$ colors can be obtained like so: every edge, as an edge of $π$, intersects $l$ on some point $p\_k$. Assign the edge with color $k$. Two edges with the same color are disjoint in $H$, otherwise they would share two vertices in $π$: one in $H$, the other some $p\_k$, contradiction.


As $n+1$ is $o(n^2)$, and projective planes with arbitrary large orders exist, $χ\_e(H)<1/k⋅|V|$ is surely satisfied.


