Bowler–Emde–Gut question on Maker–Breaker games on infinite graphs

Let $B=K_{\aleph_0}$ , and let $G_1,\dots,G_k\subseteq B$ be finitely many infinite subgraphs. In the color-preserving Maker–Breaker game on $B$ , Maker and Breaker alternately claim previously unclaimed edges of $B$ . Determine, for arbitrary finite $k$ and arbitrary choices of $G_1,\dots,G_k$ , which player has a winning strategy, where Maker wins if the edges claimed by Maker contain a copy $K\cong K_{\aleph_0}$ such that $K$ contains infinitely many edges of every $G_i$ , that is, $\lvert E(K)\cap E(G_i)\rvert=\aleph_0$ for every $i\in\{1,\dots,k\}$ . More broadly, determine the winner in the corresponding partially pattern-preserving games, in which Maker must claim a copy $K\cong K_{\aleph_0}$ such that $G_i\cap K$ is isomorphic to a subgraph of $G_i$ for every $i$ .

References

Primary source

arXiv

Additional references

Maker-Breaker games on infinite graphs with precolored edges — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new paper settles the one-pattern case and adds partial results for more patterns, but the full infinite-graph question remains open.

The Bowler–Emde–Gut question concerns which player can force prescribed colored structures in Maker–Breaker games on infinite graphs. The latest work settles the single-pattern case while leaving the general problem open.

Known results

Bowler, Emde, and Gut characterized the vertex-coloring game for finitely many colors, while showing limitations when infinitely many colors are allowed.
Related games on uncountable infinite graphs have outcomes depending on set-theoretic assumptions, including CH and MA.

August 24, 2026 partial characterization

A new paper proves Maker and Breaker conditions for $k>1$ and completely characterizes the $k=1$ color-preserving game. It therefore resolves the single-pattern case, but gives no complete characterization for all $k>1$ or all pattern-preserving games; this reported advance is unverified.

Current status (as of August 2026): The $k=1$ color-preserving case is claimed characterized, and conditions are claimed for $k>1$ , but the broader question remains open and the new results are unverified.

Sources

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