Michael–Traves conjecture: the Roller Coaster Conjecture

Let $q$ be a positive integer, let $\pi$ be a permutation of $\{\lceil q/2\rceil,\lceil q/2\rceil+1,\ldots,q\}$, and let $i_k(G)$ denote the number of independent sets of cardinality $k$ in a graph $G$. A graph is well-covered if all its maximal independent sets have the same size, and $\alpha(G)$ denotes its independence number. Roller Coaster Conjecture. There is a well-covered graph $G$ with $\alpha(G)=q$ such that

$$i_{\pi(\lceil q/2\rceil)}(G)<i_{\pi(\lceil q/2\rceil+1)}(G)<\cdots<i_{\pi(q)}(G).$$

The paper proves this conjecture by constructing suitable well-covered graphs, so the asserted ordering is no longer open.