On the minimum number of edges in triangle-free 5-critical graphs

We prove that there exists ϵ,δ>0 such that if G is a 5-critical graph, then |E(G)|≥(94+ϵ)|V(G)|−54−δT(G) where T(G) is the maximum number of vertex-disjoint cliques of size three or four where cliques of size four have twice the weight of a clique of size three. As a corollary, a triangle-free 5-critical graph G satisfies: |E(G)|≥(94+ϵ)|V(G)|−54.