Online size Ramsey numbers: Odd cycles vs connected graphs

Given two graph families H 1 and H 2 , a size Ramsey game is played on the edge set of K N . 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 H 1 or a blue copy of a graph from H 2 as soon as possible. The online (size) Ramsey number r ( H 1 , H 2 ) is the smallest number of rounds in the game provided Builder and Painter play optimally. We prove that if H 1 is the family of all odd cycles and H 2 is the family of all connected graphs on n vertices and m edges, then r ( H 1 , H 2 ) phi n n + m - 2 phi phi + 1, where phi is the golden ratio, and for n 3, m (n n - 1)2/4 2 / 4 we have r ( H 1 , H 2 ) n + 2m m + O ( root m - n + 1). We also show that r(C3, ( C 3 , P n ) 3n-4 n- 4 for n 3. Asa consequence we get 2.6n-3 . 6 n- 3 ( C 3 , P n ) 3n-4 n- 4 for every n 3.