Worked solution: Gladkov–Pak–Zimin's explicit counterexample disproving the bunkbed conjecture (2024)

Step 3 of 8: The obstruction: no graph gadget can exactly copy a hyperedge

In plain words

The obvious next move is to try to replace each triangular hyperedge of Hollom's hypergraph with a small cluster of ordinary edges — a "gadget" — that behaves exactly like a hyperedge under ordinary bond percolation: connect all three attachment points together, or none of them, with the right probabilities, and never connect just two.

But this turns out to be impossible in principle. Ordinary edges fail or survive one at a time, so any gadget necessarily has some intermediate states where exactly two of the three attachment points get connected while the third is cut off — an outcome a genuine hyperedge (all three or none) never produces. No amount of cleverness in the gadget's design can close off this leak entirely.

\nexists \ \text{graph gadget exactly simulating a single 3-hyperedge under bond percolation}

Detailed analysis

In the WZ hypergraph percolation model of Wierman and Ziff (2011), a hyperedge $e = \{a,b,c\}$ resolves into one of five outcomes, each with its own probability: all three connected ( $p_{abc}$ ), none connected ( $p_{a|b|c}$ ), only $b$ and $c$ connected while $a$ is cut off ( $p_{a|bc}$ ), or $a$ connected to exactly one of $b, c$ with the other cut off ( $p_{ab|c}$ or $p_{ac|b}$ ). A genuine $3$ -uniform hyperedge under ordinary hypergraph percolation has $p_{a|bc} = 0$ whenever $a$ is the transversal vertex of the hyperedge (the transversal vertex is either connected to both others or to neither, never to just one).

Gladkov and Zimin (2024, cited as GZ24 in Gladkov, Pak & Zimin 2024, Thm 1.5) proved that this exact outcome distribution — in particular, $p_{a|bc} = 0$ together with the other four probabilities matching a genuine hyperedge exactly — can never be reproduced by any finite graph gadget under ordinary bond percolation, however the gadget's edges and their individual retention probabilities are chosen. Intuitively: ordinary edges fail independently one at a time, so some sequence of edge failures inside any gadget must eventually separate exactly one of $a, b, c$ while leaving the other two connected, giving $p_{a|bc} > 0$ — impossible to avoid entirely.

This is precisely why Hollom's hypergraph counterexample, however striking, cannot immediately be converted into a graph counterexample by naive substitution: the mismatch at $p_{a|bc}$ means any such substitution introduces a small amount of exactly the 'wrong kind' of connectivity event that a true hyperedge would forbid. The next step shows this obstruction can be worked around, not eliminated.

Knowledge used in this step

Events and probability

Common mistake. It might seem that using more and more edges in the gadget, or tuning probabilities ever more finely, should eventually drive

p_{a|bc}

down to exactly zero in the limit; Gladkov and Zimin's theorem rules this out completely — no finite graph gadget achieves

p_{a|bc} = 0

exactly, so the mismatch, however small, is unavoidable in principle and must be controlled rather than eliminated.