On the degree, size, and chromatic index of a uniform hypergraph

Let ℋ be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the maximum degree of a vertex of ℋ. We conjecture that for every ε > 0, if D is sufficiently large as a function of t, k, and ε, then the chromatic index of ℋ is at most (t - 1 + 1/t + ε) D. We prove this conjecture for the special case of intersecting hypergraphs in the following stronger form: If ℋ is an intersecting k-uniform hypergraph in which no two edges share more than t common vertices and D is the maximum degree of a vertex of ℋ, where D is sufficiently large as a function of k, then ℋ has at most (t - 1 + 1/t) D edges.