**Motivation.** The Euro 2024 ~~soccer~~ football championship is in full swing, and the male part of my family are avid watchers. Right now the championship is in the group stage where every group member plays against every other member exactly once. I got the impression that the games in the groups get more attractive over time.

More precisely, suppose we have 3 Teams $A, B, C$, and the "attractivity scores" are $s(A) = 0, s(B) = 1, s(C) = 2$. (A higher score indicates a "more attractive" team.) The attractivity of a game is determined by the *sum of the attractivity* of the two teams partaking. So $\newcommand{\attr}{\text{attr}}\attr(\{A,B\}) = s(A) + s(B) = 1$, and $\attr(\{A,C\}) = 2$, and finally $\attr(\{A,C\}) = 3$. Because the "game attractivities" $\attr(\cdot)$ are all distinct, it makes it easy to arrange the games so that the games get more and more attractive over time.

**Formalization and generalization.** For any set $X$, let $[X]^2=\big\{\{x,y\}:x\neq y \in X\big\}$. Let $\newcommand{\N}{\mathbb{N}}\N=\{0,1,\ldots,\}$ and let $s:\N\to\N$ be a function (corresponding to the scoring function from the motivational part of this post). To $s$ we associate a map $\attr_s:[\N]^2\to [\N]$ with
$$p\in[\N]^2 \mapsto s\big(\min(p)\big) + s\big(\max(p)\big).$$
We call the map $s:\N\to\N$ *good* if $\attr_s$ is injective. An example of a good map is $s:\N\to\N$ defined by $n\mapsto 2^n$.

**Questions.**

Is there a good map $s:\N\to\N$ such that $\text{im}(\attr_s) = \N\setminus\{0\}$?

Is there a good map $s_0:\N\to\N$ such that for every good map $s:\N\to\N$ and all $n\in\N$ we have $s_0(n) \leq s(n)$?