Breaking Points in Diameter Positional Games

Definitions and mathematical statement

Consider the biased diameter games $D_d(a:b)$ on the edges of $K_n$ in the report. Under the usual Maker--Breaker interpretation, Maker and Breaker alternately claim previously unclaimed edges, $a$ and $b$ per turn respectively, and Maker wins if Maker's graph has diameter at most $d$. Graph distance in a disconnected graph is infinite. The source does not state turn order, end-of-board rules, or confirm the exact winning condition, so these conventions must be checked.

With the usual monotone bias convention, a breaking point may be denoted \[ b_{d,a}(n)=\min\{b\ge1:\text{Breaker has a winning strategy in }D_d(a:b)\}. \] The report's precise threshold convention must be recovered. The general conjectural request is that this breaking point be asymptotically close, up to a polylogarithmic factor, to the known Breaker-winning bias bounds for the parameter pairs considered in the report. Those bounds and the range of $(d,a)$ are not supplied, so no explicit general comparison function is invented.

The restored statement singles out two tasks:

1. Is the breaking point for $D_3(3:b)$ of order $n^{2/3}\operatorname{polylog}(n)$?
2. Determine the breaking point for $D_3(2:b)$.

Here the unknown variable is Breaker's bias $b$ as $n\to\infty$, while $d=3$ and Maker's biases are fixed as stated. The source gives no candidate exponent for the second case.

The notation polylogarithmic is not precise enough to specify a single asymptotic equivalent: it may mean an unspecified power of $\log n$, or an allowed multiplicative loss above and below a reference bound. Its exponent, sign, constants, and the direction of the known comparison must be recovered. An assertion merely of the form $n^{2/3+o(1)}$ would not by itself preserve a fixed polylogarithmic-factor requirement. No exact logarithmic order or leading constant is silently chosen.

Short English statement

The question asks whether the bias threshold in the report's diameter games matches known Breaker-winning bounds up to logarithmic powers, especially the $n^{2/3}$ scale for Maker bias three. It separately asks for Maker bias two, without supplying a predicted scale.

Sources

• Local UnsolvedMath registry, permanent ID 30000735, code catalogue entry, "Breaking Points in Diameter Positional Games." Proposed year 2007; proposer unspecified.
• Mini-Workshop: Positional Games (2008), Oberwolfach Reports, pp. 1073--1114, DOI 10.4171/owr/2007/20; https://doi.org/10.4171/owr/2007/20. The displayed report year 2008 differs from the DOI and proposed year 2007. This is the sole supplied literature reference.
• Statement status corrected_verified; the supplied verification says official-report content was checked and wording/notation normalized without changing the source-level claim. The retained original only states the general comparison; the explicit two cases occur in the restored field.
• Literature triage checked 2026-08-21: open. It calls the break-point comparison conjectural and says no definitive resolution of this extracted claim was located. The exact game rules, bounds, and current status require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.