Equality of List and Ordinary Ramsey Numbers

Definitions and mathematical statement

For integers $n\geq 1$, let $K_n$ denote the finite simple complete graph on $n$ vertices. For $t\geq 2$, a monochromatic copy of $K_t$ in an edge-coloured graph is a set of $t$ vertices whose pairs are all edges of one common colour. For an integer $r\geq 1$, write $[r]=\{1,\ldots,r\}$. The ordinary Ramsey number is \[ R(K_t,r)=\min\left\{n\geq 1:\text{every }c:E(K_n)\to[r]\text{ contains a monochromatic copy of }K_t\right\}. \] An $r$-list assignment on $E(K_n)$ assigns to each edge $e$ a set $L(e)\subseteq\{1,2,\ldots\}$ of exactly $r$ colour labels. An $L$-colouring is any function $c:E(K_n)\to\{1,2,\ldots\}$ satisfying $c(e)\in L(e)$ for every edge $e$. No proper-colouring condition is imposed. Under the standard list-Ramsey interpretation used in this draft, \[ R_L(K_t,r)=\min\left\{n\geq 1:\text{there exists an }r\text{-list assignment }L\text{ on }E(K_n)\text{ such that every }L\text{-colouring contains a monochromatic copy of }K_t\right\}. \] Thus the list assignment is existentially quantified, and the colourings selected from that assignment are universally quantified; lists on different edges may differ.

The two supplied questions, on the conventional parameter range above, are whether \[ R_L(K_k,2)=R(K_k,2)\qquad\text{for every integer }k\geq 2, \] and whether \[ R_L(K_3,r)=R(K_3,r)\qquad\text{for every integer }r\geq 1. \] The supplied record does not explicitly define the list-Ramsey convention or state the lower bounds on its parameters. The definition and conventional ranges displayed here therefore require confirmation against the source; the record does not resolve whether any degenerate clique or zero-colour conventions are intended. Its open-status label and its claim of prior statement verification are supplied curation, not independent mathematical verification by this draft.

Short English statement

Does the list Ramsey number equal the ordinary Ramsey number for every clique with two colours, and for triangles with any positive number of colours? Here the list Ramsey number uses an assignment of colour lists that forces a monochromatic clique in every colouring chosen from those lists. The precise convention and omitted parameter bounds require source confirmation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004136$, in the Oberwolfach Reports Open Problems collection; exact source problem code catalogue entry. No individual proposer is supplied. The recorded proposed year is $2019$.
• Supplied report citation: Combinatorics, Probability and Computing (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/19, pp. 1113--1189. https://doi.org/10.4171/owr/2019/19. The background describes these reports as records of mathematical workshops and their open-problem discussions. The citation year $2020$, the DOI containing $2019$, and the recorded proposed year $2019$ are retained as supplied.
• Supplied literature reference: List Ramsey numbers. https://arxiv.org/abs/1902.07018.
• Supplied literature reference: Journal of Graph Theory version. https://doi.org/10.1002/jgt.22610.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the list-Ramsey literature provides bounds and examples, while the two displayed equality questions remain undecided, including the relevant clique cases. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified. Its verification note reports that the official Oberwolfach report was checked directly and that the clean statement preserves the target while restoring definitions and removing source-dependent wording. This is a supplied verification claim; no source has been independently visited or checked for this draft.