Online Size Ramsey Numbers: Odd Cycles vs Connected Graphs

Given two graph families H₁ and H₂, a size Ramsey game is played on the edge set of KN. In every round, Builder selects an edge and Painter colours it red or blue. Builder is trying to force Painter to create a red copy of a graph from H₁ or a blue copy of a graph from H₂ as soon as possible. The online (size) Ramsey number r (H₁, H₂) is the smallest number of rounds in the game provided Builder and Painter play optimally. We prove that if H₁ is the family of all odd cycles and H₂ is the family of all connected graphs on n vertices and m edges, then r (H₁, H₂) n + m-2+1, where is the golden ratio, and for n 3, m (n-1) ²/4 we have r (H₁, H₂) n+2m+O (m-n+1). We also show that r (C₃, Pₙ) 3n-4 for n 3. As a consequence we get 2. 6n-3 r (C₃, Pₙ) 3n-4 for every n 3.