Abstract
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 $\frac{n-t+1}{t+1}$ 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.
Highlights
For families of graphs G1, . . . , Gt, the Ramsey number, R(G1, . . . , Gt), is the least integer n such that any t-coloring of E(Kn) contains a copy of some Gi ∈ Gi in color i for some i ∈ [t]
Decades of study have spawned a myriad of generalizations and variants of the Ramsey numbers
We consider a variant called the restricted online Ramsey numbers, which is defined through a two-player game between players Builder and Painter
Summary
For families of graphs G1, . . . , Gt, the Ramsey number, R(G1, . . . , Gt), is the least integer n such that any t-coloring of E(Kn) contains a copy of some Gi ∈ Gi in color i for some i ∈ [t]. We consider a variant called the restricted online Ramsey numbers, which is defined through a two-player game between players Builder and Painter. Gt; n) to be the smallest for which Builder can guarantee to win the locating game on Kn by querying at most edges This is an immediate extension of the original definition of R since Case 2 can never occur if n R(G1, . In this game, Painter is required to guarantee the existence of one of the monochromatic graphs and Builder must determine which color contains it. Builder has won the game if she has determined that Painter’s coloring must contain one of the monochromatic graphs in color c.
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