Uniform Winning Times in Strong Ramsey Games

Definitions and mathematical statement

For integers $n\geq k\geq 2$, let $K_n$ denote the complete graph on $n$ vertices. Under the standard interpretation of the strong Ramsey game $R(K_n,K_k)$, two players alternately claim one previously unclaimed edge of $K_n$, with the first player moving first. A player wins immediately upon claiming every edge of a copy of $K_k$. The game is drawn if all edges have been claimed without either player winning.

Question. Is it true that for every fixed integer $k\geq 2$ there exist integers $T(k)\geq 1$ and $N(k)\geq k$ such that, for every integer $n\geq N(k)$, the first player has a strategy in $R(K_n,K_k)$ that guarantees a win on or before the first player's $T(k)$-th move, regardless of the second player's legal choices? In particular, the strategy must prevent the second player from winning earlier. The strategy may depend on $n$ and $k$, but the bound $T(k)$ must be independent of $n$.

Interpretive qualifications. The supplied statement names the strong Ramsey game without explicitly giving its rules or the range of $k$; the formulation above uses the standard edge-claiming interpretation and the range $k\geq 2$. The phrase "as $n\to\infty$" is interpreted as requiring the assertion for every sufficiently large $n$. Moves are counted as moves of the first player; counting both players' turns instead gives the same boundedness question, since a win on the first player's $t$-th move occurs on turn $2t-1$. These conventions require confirmation against the cited source.

Short English statement

For each fixed clique size $k$, can the first player force a win in the strong edge-claiming game on $K_n$ within a number of moves depending only on $k$, for every sufficiently large $n$? The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30003970$, title Uniform Winning Times in Strong Ramsey Games, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection.
• Supplied primary-source citation: Mini-Workshop: Positional Games (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/44, pp. 2715--2738. https://doi.org/10.4171/owr/2018/44. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Strong Ramsey games. https://www.sciencedirect.com/science/article/pii/S0195669820300172. No authors or further bibliographic details are supplied in the record.
• Supplied literature assessment, checked on exactly 2026-08-21: the record states that the cited primary literature still describes Beck's bounded-time complete-graph question as open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record requests independent verification of time-sensitive status and cited claims.
• Attribution and dates: the supplied assessment associates the question with Beck, while the explicit proposer field is empty. The record gives proposed year 2018 and the tag owr-year:2018, whereas its report citation is labeled 2019. These supplied date labels are retained without resolving their relationship.
• Supplied editorial labels: problem status open and statement status corrected_verified. The record reports manual semantic, typographic, and LaTeX review of the clean statement. These are source-record assertions and do not constitute independent mathematical or source verification of this draft.