A shorter version of this question was posted on Math Stack Exchange.
Let $V$ be a nonempty set. $(V,S)$ is a graph if $S\subseteq\binom V2,$ a triple system if $S\subseteq\binom V3,$
a quadruple system if $S\subseteq\binom V4.$
Definition. For lack of a better word, I will say that a triple system $(V,T)$ is nice if it satisfies any of the following conditions, which are easily seen to be equivalent:
- $\left|\binom X3\cap T\right|$ is even for every $X\in\binom V4;$
- there is a point $v\in V$ such that $\left|\binom X3\cap T\right|$ is even whenever $v\in X\in\binom V4;$
- there is a set $E\subseteq\binom V2$ such that $T=\left\{t\in\binom V3:\left|\binom t2\cap E\right|\text{ is odd}\right\};$
- there is a set $E\subseteq\binom V2$ such that $T=\left\{t\in\binom V3:\left|\binom t2\cap E\right|\text{ is even}\right\}.$
Definition. $f(n)$ is the least number $m$ such that, given any triple system $(V,T)$ with $|V|=m,$ we can find an $n$-element set $W\subseteq V$ such that the triple system $\left(W,T\cap\binom W3\right)$ is nice.
(In other words, for any $4$-element set $X\subseteq W,$ either all or none or half of the four $3$-element subsets of $X$ belong to $T.$)
Question 1. Is there any literature on $f(n)$?
Question 2. What is $f(5)$?
Here are some trivial bounds for $f(n)$ in terms of ordinary Ramsey numbers.
Definition. The Ramsey number $R(n_1,n_2;d)$ is the least number $m$ such that, given an
$m$-element set $V$ and any set $S\subseteq\binom Vd,$ we can find a set $W\subseteq V$ such that either $|W|=n_1$ and $\binom Wd\cap S=\emptyset,$ or else $|W|=n_2$ and $\binom Wd\subseteq S.$
Definition. $g(n)$ is the least number $m$ such that, given any quadruple system $(V,Q)$ with $|V|=m,$ either we can find a $5$-element set $U\subseteq V$ such that $\left|\binom U4\cap Q\right|$ is even, or else we can find an $n$-element set $W\subseteq V$ and a point $v\in W$ such that $\left\{X:v\in X\in\binom W4\right\}\subseteq Q.$
Definition. $ES(n)$ is the least number $m$ such that any set of $m$ points in the plane, no three of which are collinear, contains the vertices of a convex $n$-gon.
Easy upper bounds:
$f(n)\le R(n,n;3).$
$f(4)=5\lt R(4,4;3)=13.$
$f(n)\le g(n)\le R(5,n;4).$
Easy lower bounds
$f(n)\ge ES(n)\ge2^{n-2}+1$ for $n\ge2.$
$f(R(m,n;2)+1)\ge R(m+1,n+1;3),$
in fact,
$f(R(m,n;2)+1)\ge f(h(m+1,n+1))\ge R(m+1,n+1;3),$
where $h(m+1,n+1)$ is defined in my other question Another funny kind of Ramsey number.
