Even-path inverse Turán conjecture

Let $t$ be a positive integer, let $P_{2t}$ denote the path of length $2t$, and let $\operatorname{ex}^{-1}(k,H)$ denote the inverse Turán number of $H$, namely the maximum number of edges in a graph whose every $H$-free subgraph has fewer than $k$ edges. A complete $t$-partite graph is balanced when its $t$ vertex classes have equal sizes. Even-path inverse Turán conjecture. The inverse Turán number of a path of length $2t$ is attained asymptotically by a balanced, complete $t$-partite graph. Therefore, for every $t$,

$$\operatorname{ex}^{-1}(k,P_{2t})=\frac{k^2}{2(t-1)^2}\left(1-\frac{1}{t}\right)+o(k^2).$$

This conjecture identifies balanced complete multipartite graphs as asymptotically extremal for even paths; the source presents it as an open question.