Source: USATST-2017-P1
Category: Extremal Problems

Problem:
In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is *color-identifiable* if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned *any* signature color of a different team in $S$. For all positive integers $n$ and $t$, determine the maximum integer $g(n,t)$ such that: In any sports league with exactly $n$ distinct colors present over all teams, one can always find a color-identifiable set of size at least $g(n,t)$.

Solution:
The answer is $\lceil n/t \rceil$.

To see that $\lceil n/t \rceil$ is an upper bound, note that one can easily construct a sports league with exactly $\lceil n/t \rceil$ teams in total. (Since the total number of teams in the league is $\lceil n/t \rceil$, no subset of teams can be strictly larger than this.)

A quick warning: It is common to misread the definition of color-identifiable by ignoring the word "any". Here is an illustration. Suppose we have two teams, MIT and Harvard; the colors of MIT are red, gray, and black, and the colors of Harvard are red and white. (Thus $n=4$ and $t=3$.) The assignment of MIT to gray and Harvard to red is *not* acceptable because red is a signature color of MIT, even though it is not the one assigned to MIT.

We present two proofs of the lower bound.

**Approach by deleting teams (Gopal Goel):**
Initially, place all teams in a set $S$. Then we repeat the following algorithm: If there is a team all of whose signature colors are shared by some other team in $S$ already, then we delete that team. (If there is more than one such team, we pick one arbitrarily.) At the end of the process, all $n$ colors are still present at least once, so at least $\lceil n/t \rceil$ teams remain. Moreover, since the algorithm is no longer possible, the remaining set $S$ is already color-identifiable.

It might seem counter-intuitive that we are *deleting* teams from the full set when the original problem is trying to get a large set $S$. This is less strange when one thinks of it instead as "safely deleting useless teams". Basically, if one deletes such a team, the problem statement implies that the task must still be possible, since $g(n,t)$ does not depend on the number of teams: $n$ is the number of colors present, and deleting a useless team does not change this. It turns out that this optimization is already enough to solve the problem.

**Approach by adding colors:**
For a constructive algorithmic approach, the idea is to greedily pick by color (rather than by team), taking at each step the least used color. Select the color $C_1$ with the *fewest* teams using it, and a team $T_1$ using it. Then delete all colors $T_1$ uses, and all teams which use $C_1$. Note that by the problem condition, this deletes at most $t$ colors total. Any remaining color $C$ still has at least one user. Indeed, if not, then $C$ had the same set of teams as $C_1$ did (by the minimality of $C$), but then it should have been deleted as a color of $T_1$. Now repeat this algorithm with $C_2$ and $T_2$, and so on. This operation uses at most $t$ colors each time, so we select at least $\lceil n/t \rceil$ colors, corresponding to at least $\lceil n/t \rceil$ teams.

A greedy approach by team *does not work*. For example, suppose we try to "grab teams until no more can be added". As before, assume our league has teams MIT and Harvard; the colors of MIT are red, gray, and black, and the colors of Harvard are red and white. (Thus $n=4$ and $t=3$.) If we start by selecting MIT and red, then it is impossible to select any more teams; but $g(n,t) = 2$.

Answer: the ceiling of $n/t$

Grading guidelines:
(Partial) 1. Gave a correct extremal construction showing g(n,t) \le \lceil n/t\rceil (e.g. a league with exactly \lceil n/t\rceil teams using all n colors, each team using at most t colors). 2. Made the key observation that in any color-identifiable set S, each chosen team must have an assigned color that appears in no other team of S (equivalently, the assigned colors are pairwise disjoint from all other teams’ color-sets). 3. Presented a valid deletion or greedy framework (e.g. iteratively delete a team all of whose colors are shared with other remaining teams, or greedily select a least-used color and remove related colors/teams), even if the proof that it produces at least \lceil n/t\rceil teams is incomplete. (Almost) 1. Proved the lower bound \ge \lceil n/t\rceil using the deletion algorithm, but did not fully justify one local step (e.g. why the process stops with a color-identifiable set, or why all n colors remain present so the remaining number of teams is at least \lceil n/t\rceil). 2. Proved the lower bound via the color-greedy algorithm and correctly argued ‘at most t colors are removed per step’, but left a gap in the minimality argument that every remaining color still has a user, or did not clearly deduce that the selected teams can be assigned distinct identifying colors. 3. Established both bounds and stated g(n,t)=\lceil n/t\rceil, but made a minor definitional slip (typically mis-handling the word “any” in ‘no team is assigned any signature color of a different team’) in an example or sub-argument while the main proof structure is otherwise correct.