Motivation. A while ago I attended a party and I only knew some, but not all, of the attendees. There were 2 kinds of drinks: beer and soda. I noticed that amongst my acquaintances, more than half drank beer. So in order to go against what I perceived as the mainstream ("drink a beer"), I picked a soda. Then I wondered whether a beer / soda mapping to all the guests is possible so that everyone has a drink such that the larger part of his acquaintances have the other drink.

Let's formalize this.

Formal version. Let $G=(V,E)$ be a simple, undirected graph (not necessarily finite). For $v\in V$, let $N(v) = \{w\in V: \{v,w\}\in E\}.$

If $\kappa > 0$ is a cardinal, we call a map $d: V \to \kappa$ a "drinking map" if for all $v\in V$ with $N(v) \neq \emptyset$ we have $$|N(v)\cap d^{-1}(\{d(v)\})| \leq |N(v) \setminus d^{-1}(\{d(v)\})|.$$

(We imagine $d(v)$ to be the "drink" that $v$ is having, and $v$ does not want that more than half of her friends are having $d(v)$ as well.)

Let the "drinking number" of $G$ be the smallest cardinal $\kappa$ such that there is a "drinking map" $d: V\to \kappa$, and denote it by $\text{dr}(G)$.

Question. Can $\text{dr}(G)$ become arbitrarily large for finite graphs? How about infinite graphs?

Motivation. A while ago I attended a party and I only knew some, but not all, of the attendees. There were 2 kinds of drinks: beer and soda. I noticed that amongst my acquaintances, more than half drank beer. So in order to go against what I perceived as the mainstream ("drink a beer"), I picked a soda. Then I wondered whether a beer / soda mapping to all the guests is possible so that everyone has a drink such that at least half of his acquaintances have the other drink.

Let's formalize this.

Formal version. Let $G=(V,E)$ be a simple, undirected graph (not necessarily finite). For $v\in V$, let $N(v) = \{w\in V: \{v,w\}\in E\}.$

If $\kappa > 0$ is a cardinal, we call a map $d: V \to \kappa$ a "drinking map" if for all $v\in V$ with $N(v) \neq \emptyset$ we have $$|N(v)\cap d^{-1}(\{d(v)\})| \leq |N(v) \setminus d^{-1}(\{d(v)\})|.$$

(We imagine $d(v)$ to be the "drink" that $v$ is having, and $v$ does not want that more than half of her friends are having $d(v)$ as well.)

Let the "drinking number" of $G$ be the smallest cardinal $\kappa$ such that there is a "drinking map" $d: V\to \kappa$, and denote it by $\text{dr}(G)$.

Question. Can $\text{dr}(G)$ become arbitrarily large for finite graphs? How about infinite graphs?