Bunkbed conjecture

Let $G = (V, E)$ be a finite graph, and let its bunkbed graph be the Cartesian product $G \,\square\, K_2$ consisting of two identical copies $G_0, G_1$ of $G$ joined by vertical edges $(v, 0)(v, 1)$ for vertices $v \in T \subseteq V$. Under independent bond percolation on $G_0$ and $G_1$ with retention probability $p \in (0, 1)$, the conjecture asserted that for every $u, v \in V$, the probability that $(u, 0)$ is connected to $(v, 0)$ in the same layer is at least the probability that $(u, 0)$ is connected to $(v, 1)$ in the opposite layer.

In October 2024, Nikita Gladkov, Igor Pak, and Aleksandr Zimin disproved the bunkbed conjecture (`arXiv:2410.02545`, published in PNAS in 2025) by constructing an explicit planar counterexample graph with $7,222$ vertices and $14,442$ edges, building on Lawrence Hollom's 2024 counterexample for hypergraph percolation (`arXiv:2406.01790`). Their proof is purely analytical and does not rely on brute-force computer verification.