Caro's linear odd induced subgraph conjecture

Let $G$ be a finite simple graph on $n$ vertices with no isolated vertices. An odd induced subgraph is an induced subgraph in which every vertex has odd degree, and let $f_o(G)$ denote its maximum order.

Caro's conjecture. There exists a positive constant $c$ such that every $n$-vertex graph without isolated vertices contains an odd induced subgraph with at least $cn$ vertices.

This conjecture was posed in connection with the problem of finding large induced subgraphs whose vertices all have odd degree. It was proved by Ferber and Krivelevich with the absolute choice $c=10^{-4}$, so the conjecture is solved.