Let $D \subset R$ and let $f: D\rightarrow R.$ Assume that #D $\geq 4$. Assume that $f$ is strictly 4-monotone, i.e., assume, for all $S\subset D, $ that [#$S=4] \rightarrow [(f|S)$ is strictly monotone]. Prove that f is strictly monotone.
I have been able to prove monotone included problems, but i don't know how to involve the cardinality portion of this question.
Let's see. We have to prove that, if a function is strictly monotone on every $4$-point set, then it is strictly monotone. In other words, if $f$ fails to be strictly monotone, then it fails to be strictly monotone on some $4$-point subset of its domain.
Hmm. What does it mean for $f$ to be strictly monotone? It means that $f$ is either strictly increasing or strictly decreasing. So, if $f$ is not strictly monotone, then it's neither strictly increasing nor strictly decreasing.
If $f$ is not strictly increasing, that means there are two points $a,b\in D$ such that $a\lt b$ while $f(a)\ge f(b).$
If $f$ is not strictly decreasing, that means there are two points $c,d\in D$ such that $c\lt d$ while $f(c)\le f(d).$
Putting that together, if $f$ is not strictly monotone, then there are points $a,b,c,d\in D$ such that $a\lt b, f(a)\ge f(b), c\lt d, f(c)\le f(d).$ So $f$ is not strictly monotone on the set $\{a,b,c,d\}$!
Are we done? No, because $\{a,b,c,d\}$ is not necessarily a $4$-point set; there could be some overlap between $\{a,b\}$ and $\{c,d\}.$ Can we add points to $\{a,b,c,d\}$ to get a $4$-point set? Yes, because there are at least $4$ points in $D.$ Done!
Proof. Assume for a contradiction that $f$ is not strictly monotone. Since $f$ is not strictly increasing, we can choose $a,b\in D$ so that $f(a)\ge f(b).$ Since $f$ is not strictly decreasing, we can choose $c,d\in D$ so that $c\lt d$ and $f(c)\le f(d).$ Since $D$ has at least $4$ elements, we can find a set $S$ containing exactly $4$ elements such that $a,b,c,d\in S\subseteq D.$ Then $f$ is not strictly monotone on $S$ and we have arrived at a contradiction.
Hmm. Maybe, with a bit more work, we can prove that $f$ is strictly monotone if it's strictly monotone on every $3$-point subset of its domain.