The extremal functions for triangle-free graphs with excluded minors.

Abstract 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 | E (G) | ≤ 3 | V (G) | − 9 + t ∕ 3. 2. For p = 2 , 3 , ... , 9 , a triangle-free graph G on at least 2 p − 5 vertices with no K p -minor satisfies | E (G) | ≤ (p − 2) | V (G) | − (p − 2) 2. [ABSTRACT FROM AUTHOR]