A conjecture of Berge about linear hypergraphs and Steiner systems \(S(2,4,v)\) (Q1849887)

This paper studies a conjecture of Berge about linear hypergraphs and Steiner systems \(S(2,4,v)\). A hypergraph is a pair \(H= (X,C)\), where \(X\) is a finite non-empty set and \(C\) is a collection of sets \(E\subseteq X\) (\(E\neq\emptyset\), for every \(E\subseteq C\)), and \(\bigcup_{E\in C} E= X\). The elements of \(X\) are called vertices of \(H\), and the edges of \(H\) are the elements of \(C\). \(H\) is called \(k\)-uniform if every edge has \(k\) distinct elements, and for \(k=2\) \(H\) is called a graph. \(H\) is said to be a linear hypergraph if any two of its edges contain at most one vertex in common. The heredity of \(H\), denoted by \(H^*\), is the hypergraph with the same point-set as \(H\) but its edge-set is the family of all non-empty sets of edges of \(H\). This paper is concerned with the following famous conjecture due to Berge: If \(H\) is a linear hypergraph, then \(H^*\) has always the edge-coloring property. Since all Steiner systems \(S(2,k,v)\) are linear hypergraphs, the study of the conjecture for such systems is very appropriate. It is already shown that all resolvable \(S(2,4,v)\) systems verify Berge's conjecture. In this paper the following result is proved: All nearly resolvable \(S(2,4,v)\) and all almost nearly resolvable \(S(2,4,v)\) verify Berge's conjecture.