Part of a problem that I'm trying to solve involves the following situation: let $S = \{t_1,...,t_n\}$ be a set of n points in a line. Let $W = \{t_{i_1},...,t_{i_k}\}$ be a k-subset of S, such that the number of odd contiguous sets of W is less than some number $(d-k)$, where d is fixed. Let $\pi = (1 2 . . . n)$ be the cyclic permutation that sends $i \to i +1$ for $1 ≤ i ≤ n − 1$, and $n \to 1$.
Show that the set $W' = \{t_{\pi{(i_1})},...,t_{\pi{(i_k})}\}$ also has at most $(d-k)$ odd contiguous sets.
In picture an example would look like this:
Here are the definitions of contiguous and end sets that we use in our class:
I'd appreciate some hints, or comment on whether some extra information is missing to argue about this situation. Thanks.
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Edit: I've realized that a bound on $(d-k)$ might be needed to argue persuasively about this situation, but I don't know how to go further. This is in fact a problem on Gale's evenness condition. Here's the full proposition and definitions:
