Intersection segments

Question

Assume $\mathcal{F}$ is a finite family of closed interval and $k\geq 2$, such that among any $k+1$ segments $I_0, I_1, \cdots, I_k\in F$ some two of them intersects. Prove there exist $k$-point set $X$ such that $$\forall I\in\mathcal{F}; I\cap X\neq\emptyset.$$

Answer 1

HINT: Let $\mathscr{F}$ be such a family for some $k\ge 2$. For $I,J\in\mathscr{F}$ write $I\prec J$ if $\max I<\min J$. Let $\mathscr{F}_0=\mathscr{F}$. Given $\mathscr{F}_\ell$ for some $\ell\ge 0$, let $\mathscr{L}_\ell$ be the set of $\prec$-minimal elements of $\mathscr{F}_\ell$, and let $\mathscr{F}_{\ell+1}=\mathscr{F}_\ell\setminus\mathscr{L}_\ell$. Let $m$ be maximal such that $\mathscr{L}_m\ne\varnothing$.

• Show that if $I\in\mathscr{L}_\ell$ for some $\ell>0$, then there is a $J\in\mathscr{L}_{\ell-1}$ such that $J\prec I$.
• Show that for $\ell=0,\ldots,m$ there is an $x_\ell\in\bigcap\mathscr{L}_\ell$.
• Show that $m\le k-1$.
• Show that $X=\{x_\ell:0\le\ell\le m\}$ has the desired property.