Is the 1/3-2/3 Conjecture for Pressing Sequences true

Definitions and mathematical statement

Let $\mathcal P(G)$ denote the finite nonempty set of successful pressing sequences of a graph $G$, with the pressing convention and permitted graph class still to be specified. A graph is uniquely pressable if $|\mathcal P(G)|=1$. Whenever the relation "$x$ occurs before $y$" is defined for every sequence in $\mathcal P(G)$, put \[ p_G(x,y)=\frac{|\{\sigma\in\mathcal P(G):x\text{ occurs before }y\text{ in }\sigma\}|}{|\mathcal P(G)|}. \] The conjecture asks whether every graph $G$ that is not uniquely pressable has two distinct vertices $x,y$ such that \[ \frac13\leq p_G(x,y)\leq\frac23. \] The random sequence is uniform on $\mathcal P(G)$, and the endpoints are included.

For context, one usual pressing model starts with a finite simple graph whose vertices are black or white. Pressing a black vertex $v$ toggles adjacency between every pair of distinct vertices in its closed neighborhood and toggles the colors of all vertices in that neighborhood. A successful sequence consists of legal presses and ends with an edgeless graph having all vertices white. This convention is not specified in the supplied item: in particular a successful sequence in this model need not contain every vertex. Thus the displayed probability is a precise formulation only after the intended sequence family and the treatment of omitted vertices have been fixed. The item also does not say whether graphs without successful sequences are excluded; uniform sampling requires a nonempty family. These issues are unresolved in the supplied source.

Short English statement

For a graph with more than one successful pressing sequence, must some pair of vertices appear in either relative order with probability at least $1/3$ under the uniform distribution on successful sequences? The pressing model, the eligibility of graphs, and the treatment of vertices omitted from a sequence are not specified in the supplied item.

Sources

• Cooper, Combinatorial Problems I Like, October 2020 snapshot, "Matroids, Lattices, and Posets," source-order bullet 47: https://people.math.sc.edu/cooper/combprob.html; accessed 2026-07-29, HTML extraction. The item refers to "See this," but the supplied record includes no destination for that link.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $3100047$, source code catalogue entry, Is the 1/3-2/3 Conjecture for Pressing Sequences true. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.
• The supplied research summary connects the question to pressing-sequence research of Cooper and collaborators and to the poset $1/3$--$2/3$ conjecture; it supplies no specific bibliographic reference for that research.