  Consider a two-player game between players Builder and Painter. Painter
begins the game by picking a coloring of the edges of $K_n$, which is hidden
from Builder. In each round, Builder points to an edge and Painter reveals its
color. Builder's goal is to locate a particular monochromatic structure in
Painter's coloring by revealing the color of as few edges as possible. The
fewest number of turns required for Builder to win this game is known as the
restricted online Ramsey number. In this paper, we consider the situation where
this "particular monochromatic structure" is a large matching or a large tree.
We show that in any $t$-coloring of $E(K_n)$, Builder can locate a
monochromatic matching on at least ${n-t+1\over t+1}$ edges by revealing at
most $O(n\log t)$ edges. We show also that in any $3$-coloring of $E(K_n)$,
Builder can locate a monochromatic tree on at least $n/2$ vertices by revealing
at most $5n$ edges.
