Let
![U=\{1,2,\ldots ,n\}](/media/m/d/7/8/d7878a3d7e544f1b98ee163b7ab23f78.png)
, where
![n\geq 3](/media/m/d/f/e/dfe037b8debb8aa67d6ed7ad5e28cc6c.png)
. A subset
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
of
![U](/media/m/d/f/a/dfa3ccb1bb2d14869d77a98d0d2baf97.png)
is said to be split by an arrangement of the elements of
![U](/media/m/d/f/a/dfa3ccb1bb2d14869d77a98d0d2baf97.png)
if an element not in
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
occurs in the arrangement somewhere between two elements of
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
. For example, 13542 splits
![\{1,2,3\}](/media/m/d/3/d/d3d0fc74e897faf89defcab673f34ab8.png)
but not
![\{3,4,5\}](/media/m/f/a/3/fa3c51db100bf6fba88b0ba96f972f0f.png)
. Prove that for any
![n-2](/media/m/e/9/c/e9c15a9049a288c79652368b3e9f2812.png)
subsets of
![U](/media/m/d/f/a/dfa3ccb1bb2d14869d77a98d0d2baf97.png)
, each containing at least 2 and at most
![n-1](/media/m/e/5/3/e5321d0e9cb5571212aaf94c7ce333b2.png)
elements, there is an arrangement of the elements of
![U](/media/m/d/f/a/dfa3ccb1bb2d14869d77a98d0d2baf97.png)
which splits all of them.
%V0
Let $U=\{1,2,\ldots ,n\}$, where $n\geq 3$. A subset $S$ of $U$ is said to be split by an arrangement of the elements of $U$ if an element not in $S$ occurs in the arrangement somewhere between two elements of $S$. For example, 13542 splits $\{1,2,3\}$ but not $\{3,4,5\}$. Prove that for any $n-2$ subsets of $U$, each containing at least 2 and at most $n-1$ elements, there is an arrangement of the elements of $U$ which splits all of them.