  In 1982, Harary introduced the concept of Ramsey achievement game on graphs.
Given a graph $F$ with no isolated vertices. Consider the following game played
on the complete graph $K_n$ by two players Alice and Bob. First, Alice colors
one of the edges of $K_n$ blue, then Bob colors a different edge red, and so
on. The first player who can complete the formation of $F$ in his color is the
winner. The minimum $n$ for which Alice has a winning strategy is the
achievement number of $F$, denoted by $a(F)$. If we replace $K_n$ in the game
by the completed bipartite graph $K_{n,n}$, we get the bipartite achievement
number, denoted by $\operatorname{ba}(F)$. In his seminal paper, Harary
proposed an open problem of determining bipartite achievement numbers for
trees. In this paper, we correct $\operatorname{ba}(mK_2)=m+1$ to $m$ and
disprove $\operatorname{ba}(K_{1,m})=2m-2$ from Erickson and Harary, and extend
their results on bipartite achievement numbers. We also find the exact values
of achievement numbers for matchings, and the exact values or upper and lower
bounds of bipartite achievement numbers on matchings, stars, and double stars.
Our upper bounds are obtained by deriving efficient winning strategies for
Alice.
