The extremal functions for triangle-free graphs with excluded minors

We prove two results: 1. A graph G on at least seven vertices with a vertex v such that G - v is planar and t triangles satisfies vertical bar E(G)vertical bar <= 3 vertical bar V(G)vertical bar - 9 + t/3. 2. For p = 2, 3, ... , 9, a triangle-free graph G on at least 2p-5 vertices with no K-p-minor satisfies vertical bar E(G)vertical bar <= (p - 2)vertical bar V(G)vertical bar - (p - 2)(2). (C) 2018 The Authors. Published by Elsevier Ltd. All rights reserved.