Bunkbed conjecture

The bunkbed conjecture says that in a randomly retained bunkbed graph, two vertices on the same layer are at least as likely to be connected as corresponding vertices on opposite layers.

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Bunkbed graph: A graph formed from two copies of a base graph, with corresponding vertices joined by vertical edges. Its two copies and vertical edges define the structure in which the conjecture is tested.
David R. Grey: A mathematician associated with formulating and studying the bunkbed conjecture. The conjecture is commonly associated with Grey's formulation and early investigation.
Trees (graph theory): Connected graphs containing no cycles. Bunkbed inequalities can be analyzed particularly directly when the base graph is a tree.
Harris–Kleitman inequality: An inequality stating that increasing events in a product probability space are positively correlated. It concerns related increasing events, but does not directly establish the bunkbed comparison.
Bond percolation: A random graph model in which each edge is independently retained with a specified probability. The conjecture compares connection probabilities after edges are independently retained.
Percolation theory: The study of random connectivity in structures whose elements are independently occupied or removed. The conjecture arose within the broader study of probabilistic connectivity.
Cycles (graph theory): Closed paths in a graph whose vertices do not repeat except at the starting endpoint. Cycle structure supplies small test cases for comparing layer-to-layer connection chances.
FKG inequality: A positive-correlation inequality for increasing functions under suitable lattice probability measures. Its powerful correlation principle does not automatically compare the two bunkbed connection events.
Graph connectivity: The property that paths join specified vertices or that a graph forms one connected component. The event whose probability the conjecture compares is the existence of a path.
Graph theory: The mathematical study of graphs, consisting of vertices and edges and their relationships. The conjecture is a graph-probability problem built from finite graphs.