This question is motivated by a real life task (which is briefly described after the question.)


Let $G=(V,E)$ be an infinite graph. For $v\in V$ let $N(v) = \{x\in V: \{v,x\}\in E\}$. If $S\subseteq V$ with $\emptyset\neq S\neq V$ we say that $v\in V$ is *happy with respect to $S$* if $$N(v)\cup \{v\}\subseteq S \text{ or }(N(v)\cup\{v\})\cap S =\emptyset.$$


We set $H(S)$ to be the collection of happy vertices with respect to $S$ and we say that $J\subseteq V$ is *maximally happy* if $J=H(S)$ for some proper nonempty subset $S\subseteq V$, and whenever $T\subseteq V$ is proper, nonempty with $J\subseteq H(T)$ then $J=H(T)$. 


**Question.** If $S$ is a proper, nonempty subset of $V$, is $H(S)$ contained in some maximally happy subset of $V$?




---


Real life motivation. I was co-organizer of a children's birthday party recently. Some of the attendants were friends of each other, others not. I was given the task of splitting the attendants into 2 teams such that for as many attendants $a$ as possible, $a$'s friends were on the same team as $a$ -- a socially tricky task.



Maximizing "happy" vertices in splitting an infinite graph
I think, no. Imagine that we have infinitely many boys $b\_1,b\_2,\dots$ and girls $g\_1,g\_2,\dots$ such that all boys are mutual friends, all girls are mutual friends and $b\_i,g\_j$ are friends if and only if $i\geqslant j$. If $b\_1$ is happy, all boys and $g\_1$ must be in the same part (without loss of generality not in $S$), therefore $S$ must contain some girl, let $m:=\min\{i:g\_i\in S\}$. Then $H(S)=\{b\_1,\dots,b\_{m-1}\}$. So, clearly there is no maximal happy set containing $b\_1$.


