Worked solution: Gladkov–Pak–Zimin's explicit counterexample disproving the bunkbed conjecture (2024)
Rather than attack ordinary graphs head-on, Lawrence Hollom looked at a natural relative of the conjecture built from hypergraphs, where a single "hyperedge" can bundle together three vertices at once instead of just two. He wrote down a small hypergraph — just vertices and triangular hyperedges — with three chosen transversal vertices, and a slightly different (but closely related) rule for how percolation crosses each level, and found that for a specific pair of vertices, connecting across levels was actually more likely than staying on the same level.
This did not settle the original conjecture about ordinary graphs, since hyperedges bundling three vertices together are a genuinely different kind of object from edges bundling only two — but it revealed exactly the kind of local imbalance that, if it could somehow be smuggled into an ordinary graph, would break the conjecture there too.
A hypergraph generalises a graph by allowing hyperedges to contain any number of vertices, not just two; a -uniform hypergraph has every hyperedge of size exactly . Lawrence Hollom (2024) studied the alternative bunkbed hypergraph percolation model, in which each hyperedge in the bottom-level hypergraph is either kept and its corresponding copy on top is deleted, or vice versa, each with probability , independently across hyperedges — a natural hypergraph analogue of ordinary bunkbed percolation.
Hollom exhibited a specific -uniform hypergraph with vertices and hyperedges, with transversal set , and computed exactly that while — a violation of the bunkbed inequality for hypergraphs, with the exact fractions small enough to check by direct enumeration of the equally likely ways the six hyperedges can resolve (Gladkov, Pak & Zimin 2024, Lemma 3.1, citing Hollom 2024).
Because hypergraph percolation with -element hyperedges is not literally bond percolation on an ordinary graph, Hollom's result is a counterexample to a natural generalisation of the bunkbed conjecture, not to the conjecture itself. But it isolates the precise combinatorial mechanism — an imbalance concentrated at vertices where several -way branchings interact — that the rest of this proof adapts to ordinary graphs.