1. In a certain sports association, each team can choose at most $t$ distinctive colors for their team uniforms. For a set $S$ of teams, if each team in $S$ can be assigned one of their distinctive colors such that no team in $S$ is assigned any distinctive color of another team in $S$, then the set $S$ is called "color-identifiable." For all positive integers $n, t$, find the maximum value of the integer $g(n, t)$ such that in any sports association where all teams' distinctive colors consist of exactly $n$ different colors, there always exists a color-identifiable set of at least $g(n, t)$ teams.
1. The maximum value of $g(n, t)$ is $\left\lceil\frac{n}{t}\right\rceil$, where $\lceil x\rceil$ denotes the smallest integer not less than the real number $x$.

Assume the set $S$ is recognizable by colors, and $|S|$ is minimized. If each color is a distinctive color of a team, and each team in $S$ has $t$ distinctive colors, then
$$
|S|=\left\lceil\frac{n}{t}\right\rceil \text {. }
$$

For an approximate algorithm, the idea is to select colors rather than teams. At each step, choose the color that is used as a distinctive color by the fewest teams.

Select color $C_{1}$, which is chosen as a distinctive color by the fewest teams. Let team $T_{1}$ choose $C_{1}$ as a distinctive color. Then delete all distinctive colors chosen by $T_{1}$ and all teams that choose $C_{1}$ as a distinctive color. By the condition, at most $t$ colors are deleted; any remaining color $C$ is still a distinctive color of some remaining team. In fact, if $C$ is not a distinctive color of any remaining team, by the minimality of $C_{1}$, the set of teams choosing $C$ and $C_{1}$ as distinctive colors is the same. Therefore, $C$ should have been deleted, leading to a contradiction.

For $C_{2}$ and $T_{2}$, etc., repeat the above algorithm. This operation deletes at most $t$ colors each time, so at least $\left\lceil\frac{n}{t}\right\rceil$ colors can be chosen.