Online Versus Ordinary Size-Ramsey Numbers

Definitions and mathematical statement

Retain the two functions named in the source: \[ N_c(t)=\mathrm{size\text{-}oRam}_c^2(t),\qquad D_c(t)=\mathrm{size\text{-}Ram}_c^2(t). \] For every fixed positive integer $c$, the proposed comparison is \[ \lim_{t\to\infty}\frac{N_c(t)}{D_c(t)}=0. \] With finite positive denominators and integer parameter $t$, its quantified form is \[ \forall c\in{\mathbb{Z}}_{>0}\ \forall\varepsilon>0\ \exists T\in{\mathbb{N}}\ \forall t\geq T,\qquad N_c(t)<\varepsilon D_c(t). \] The parameter $c$ is held fixed before the limit; this is not a uniform assertion for growing $c$.

For context, an ordinary size-Ramsey number usually minimizes the number of edges of a host graph whose every permitted coloring contains a specified monochromatic target. An online version usually minimizes the number of edges Builder must expose, against immediate irrevocable Painter color choices, to force that target. These descriptions do not define the source's displayed functions without its target family, coloring rules and parameter conventions.

The extract gives no target graph or winning property indexed by $t$, does not define the roles of $c$ and the superscript $2$, and does not state the online board, admissible moves or whether size counts edges, vertices or another resource. Thus no clique, path or chromatic-number target is silently chosen. The literal range includes $c=1$; if $c$ is intended as a color count, the one-color case needs separate checking, since under the usual unrestricted fixed-target edge-count conventions the online and ordinary costs coincide. The record's own assessment calls the extracted conjecture incomplete.

Short English statement

For each fixed positive $c$, is the online size-Ramsey quantity asymptotically negligible compared with the ordinary one? The exact target, coloring model and meanings of the parameters are missing, so only the supplied asymptotic comparison can be retained precisely.

Sources

• Supplied report: Mini-Workshop: Positional Games (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/20, pp. 1073--1114; proposed year 2007. https://doi.org/10.4171/owr/2007/20.
• Local UnsolvedMath registry, permanent ID 30000734, catalogue entry; created 25 August 2026, updated 16 September 2026. The corrected_verified note reports comparison with the official report. The supplied 21 August 2026 literature triage labels the record open, explicitly calls the extracted asymptotic conjecture incomplete, and reports no located resolution of the stated limit. No additional external reference is supplied; no independent literature assessment is asserted here.