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Maker-Breaker games on infinite graphs with precolored edges

T0 review · 2 major / 4 minor · reviewed 2026-08-28 · deepseek-v4-flash

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Pith's one-line read The paper proves a quantitative threshold for winning infinite Maker-Breaker games on precolored edges, and completely solves the two-color case.

desk verdict Strong framework and a clean two-color theorem, but the printed Maker-side bounds in Theorems 1.3–1.5 are not supported by the proof as written; Theorem 4.9's constant needs a repair. read the letter →

arxiv 2608.23349 v1 pith:K6OFMIDZ submitted 2026-08-24 math.CO

Nathan Bowler , Florian Gut , Henri Ortmüller This is my paper · ORCID
classification math.CO MSC 05C5705C6305C5591A43
keywords Maker-Breakergamesinfinitecompletegraphpositionalprecolorededgesstructure-preservingRamseytheorystarsthresholdresults
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Maker and Breaker alternately claim edges of the countably infinite complete graph $K_{\aleph_0}$, and Maker wins by claiming an infinite complete subgraph that contains infinitely many edges from each of finitely many prescribed subgraphs, or that preserves their structure. This paper establishes that, for subgraphs that become bounded after removing finitely many vertices of infinite degree, the winner is decided by a quantitative threshold: Maker wins when these 'orders' are large enough (doubly exponential in the number of subgraphs), while Breaker wins when some subfamily has total order below an explicit exponential bound. For two subgraphs the answer is complete and simple: both color classes unbounded means Maker wins, one bounded means Breaker wins. The paper also gives doubly exponential bounds on the number of disjoint infinite stars needed to force preservation of prescribed numbers of stars per color. A sympathetic reader should care because the results move infinite positional games from existence questions to a threshold picture, and because the proofs reduce the infinite game to a finite hypergraph game.

What carries the argument

The load-bearing construction is the reduction of the infinite game to the finite auxiliary game $\operatorname{MB}_{\mathrm{fin}}(C,s,t)$, played on the complete $s$-partite graph $K^s_C$ (a vertex set split into $s$ classes of size $C$): its hyperedges are the $s$-sets of pairwise crossing edges, and Maker must claim an edge set $H$ such that every coloring of the hypergraph $K^{(s)}_C(H)$ admits a color class where Maker wins one of two auxiliary vertex-claiming games. Theorems 4.5 and 4.6 prove the equivalence between this finite game and the infinite game $\operatorname{MB}_{1.4}(C,s,t)$. Inside that proof, the engine is the wish function: Maker grows a finitely branching tree whose comparability graph she claims, attaching fresh vertices whose colors satisfy the wishes of leaves, and she extracts an infinite ray from the tree; the ray's vertices then induce an infinite clique with the prescribed ascending color pattern. Breaker's wins rely on a pairing strategy over the bounded part of the board, built from a proper edge-coloring of the square of the line graph, combined with the classical threshold for the finite clique game.

What would settle it

Compute the winner of $\operatorname{MB}_{\mathrm{fin}}(C,s,t)$ for small parameters and search for a coloring of $K_{\aleph_0}$ at those parameters where the loser of the finite game wins the infinite game; any such pair refutes Theorem 4.5 or Theorem 4.6. Alternatively, construct an edge-coloring of $K_{\aleph_0}$ in which both color classes have infinite maximum degree but Breaker still prevents Maker from obtaining an infinite clique containing both colors infinitely often, which would refute Theorem 1.1.

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Extended reading notes

Core claim

The central claim, Theorem 1.5, is a dichotomy for the color-preserving game on $K_{\aleph_0}$ with $s$ graphs of finite order $C_1,\dots,C_s$ (each has exactly $C_i$ vertices of infinite degree, and bounded degree after deleting them) together with $t$ ascending graphs (containing vertices of arbitrarily large degree). If every $C_i$ is at least $\max\{2^t\cdot 2^{(1+o_s(1))3^s},\, 2^{2(1+o_s(1))s}\}$, Maker has a winning strategy. If, for some nonempty $A\subseteq[s]$, $\sum_{j\in A} C_j \le \frac{m(A)}{e} 2^{m(A)/2-1}$, where $m(A)$ is the minimum size of a vertex set meeting all vertices of infinite degree of the graphs in $A$, then Breaker has a winning strategy, and moreover his strategy forces the union of those graphs inside any infinite clique Maker claims to have order strictly less than $m(A)$. The two-color case, Theorem 1.1, is fully decided: Breaker wins exactly when one color class has finite maximum degree, otherwise Maker wins. In the partially pattern-preserving game, Theorem 1.3 shows the minimal number of pairwise disjoint copies of the infinite star $S_{\aleph_0}$ needed per color to force containment of $\ell$ stars from each of $k$ colors satisfies $2^{(1-o(1))k\ell/2}< f^{S_{\aleph_0}}_{\mathrm{par}}(k,\ell) \le 2^{2(1+o(1))k\ell}$.

Load-bearing premise

Maker's thresholds all flow through the equivalence between the infinite structure-preserving game and the finite auxiliary hypergraph game $\operatorname{MB}_{\mathrm{fin}}(C,s,t)$; if a win in the finite game did not transfer to the infinite game (or the transfer in the other direction failed), the Maker-side bounds of Theorem 1.5 would collapse.

Editorial extensions

If this is right

  • For any 2-coloring of the edges of $K_{\aleph_0}$, the winner is decided purely by whether both color classes have unbounded maximum degree: Maker wins if both are unbounded, and Breaker wins if one is bounded.
  • For several prescribed subgraphs, Maker wins whenever all orders of the non-ascending graphs lie above roughly $2^{2s(1+o_s(1))}$, yielding an explicit doubly exponential sufficient condition.
  • Breaker wins whenever a nonempty subfamily has total order at most $(m(A)/e)2^{m(A)/2-1}$, and his strategy bounds the order of the union of those graphs inside any infinite clique Maker can claim.
  • The minimum number of disjoint infinite stars needed per color to force $\ell$ preserved stars from $k$ colors lies strictly between $2^{(1-o(1))k\ell/2}$ and $2^{2(1+o(1))k\ell}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The doubly exponential gap between the Maker and Breaker thresholds suggests the true critical quantity is $m(A)$, the number of independent infinite-degree hubs, with the exponential in $m(A)$ dominating; pinning the constant would settle the three-or-more-color case.
  • The finite-hypergraph reduction is likely reusable: any infinite positional game whose winning condition is an ascending pattern along tree paths might reduce to a finite auxiliary game by the same comparability-tree construction.
  • A testable next step from the paper's own open problems is the three-color game where one color class is one or two disjoint infinite stars and another is ascending; the methods here determine it in many cases but leave a boundary case open.
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Formalized claims in Lean

  1. Claim #1: The central claim, Theorem 1.5, is a dichotomy for the color-preserving game on $K_{\aleph_0}$ with $s$ graphs of finite order $C_1,\dots,C_s$ (each has exactly $C_i$ vertices of infinite degree, and bounded degree after deleting them) together with $t$ ascending graphs (containing vertices of arbitrarily large degree). If every $C_i$ is at least $\max\{2^t\cdot 2^{(1+o_s(1))3^s},\, 2^{2(1+o_s(1))

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies Maker-Breaker games on the countably infinite complete graph K_aleph0, where Maker must claim a K_aleph0 with prescribed intersection properties relative to finitely many given subgraphs G_i. The games considered are the color-preserving game MBcol, the pattern-preserving game MBpat, and a partially pattern-preserving variant MBH_par. The main claims are: a complete two-color characterization (Theorem 1.1); bounds on the minimal number of disjoint stars needed for Maker to preserve stars, f^{S_aleph0}_par (Theorem 1.3); a general Maker sufficient condition obtained through a finite auxiliary game MBfin and an infinite-game equivalence (Theorems 1.4, 4.5, 4.6, 4.9); and a combined sufficient/necessary condition for MBcol (Theorem 1.5). The proof technique combines a wish-function construction (Section 3), a reduction to a finite hypergraph game (Section 4), Beck's threshold criterion, and a pairing strategy for Breaker using a bounded-degree edge-coloring (Section 5).

Significance. If the quantitative claims are correct, the paper makes a substantial contribution: the two-color characterization fully resolves the k=2 case of the precolored-edge question from [4], and the finite/infinite equivalence in Theorems 4.5 and 4.6 provides a concrete method for transferring finite hypergraph strategies to infinite games. The paper is largely self-contained and includes a complete proof of the wish-function theorem (Theorem 3.5), which is a strength. The Breaker-side structural guarantee in Theorem 1.5 and the explicit lower bound in Theorem 1.3 are also concrete and checkable. However, the numerical thresholds on the Maker side are not supported by the displayed proofs: the constant in Theorem 4.9 is too small for the Beck inequality used, and the asymptotic upper bounds in Theorems 1.3 and 1.5 do not follow from the application of Theorem 1.4 as written.

major comments (2)
  1. [§4.9 (proof of Theorem 4.9)] The displayed inequality used to apply Theorem 4.7, namely `2^{s-4} C^{s-3} C(s-1) < C^{s-1}/2^s - C^{s-2}`, is equivalent to `C > (s-1) 2^{2s-4} + 2^s`. The stated constant `C = ceil{s^2 2^{s-3}}` fails this inequality for every s >= 2 (for example, s=4 requires C > 64 while the statement gives C=32), and the s=1 case is degenerate for the (s-1)-uniform Beck criterion. Consequently the proof of Maker's winning strategy in MB_fin(C',s,0), and hence the t=0 case of Theorem 1.4, is invalid as written. The local inequality would hold if the first term were `s^2 2^{2s-3}`, but that is not the stated constant.
  2. [§5, Lemma 5.1 and proof of Theorem 1.3] Equation (2) in the proof of Theorem 1.3 asserts `f^{S_aleph0}_par(s,1) <= 2^{2(1+o(1))s}` as a consequence of Theorem 1.4 with t=0. However, Theorem 1.4 requires `C' >= 3C(s+1)C^{s+1}`, and with any C of order `2^{Theta(s)}` (whether the stated `s^2 2^{s-3}` or a corrected `s^2 2^{2s-3}`) this gives `C' = 2^{Theta(s^2)}`. The proof displays no argument reducing this to `2^{2(1+o(1))s}`. Since Lemma 5.1 and the Maker-side threshold of Theorem 1.5 are obtained by the same application of Theorem 1.4, the quantitative Maker side of Theorems 1.3 and 1.5 is not established as stated. The qualitative claim that sufficiently large orders suffice may survive a repair, but the printed exponents do not follow from the supplied proof.
minor comments (4)
  1. [Definition 4.4] The game MB_fin(C,s,t) is defined by saying Maker 'can claim an edge set H' with a certain property, but no stopping rule is stated; since the board is finite, the intended meaning is that the game ends when all edges of E(K^s_C) are claimed, and this should be made explicit.
  2. [Lemma 3.3] The objects `eG_i_j` are used before being formally defined; please define the disjoint-union components explicitly or replace them with a cleaner notation.
  3. [Lemma 5.1] The displayed definition `C' := min{C1_s,...,Cs_s}` is garbled by typesetting; please clarify whether the intended quantity is `min_i floor(C_i/s)` or another expression.
  4. [Theorem 1.5] The condition `C_1,...,C_s >= max{2t * 2^{(1+o_s(1))3s}, 2^{2(1+o_s(1))s}}` uses `o_s(1)` without stating the uniformity in t; please specify the intended asymptotic regime (e.g., s to infinity with t arbitrary).

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: all load-bearing tools are proved in-line or cited as standard external results; the Theorem 4.9 constant discrepancy is a correctness gap, not circularity.

full rationale

This paper proves an infinite Maker–Breaker characterization (Theorem 1.5) and a partially pattern-preserving game result (Theorem 1.4) whose Maker half flows through an explicitly proved equivalence between the infinite game MB_1.4(C,s,t) (Definition 4.1) and the auxiliary finite game MB_fin(C,s,t) (Definition 4.4). The two games are defined independently, and Theorems 4.5 and 4.6 give genuine strategies in both directions (Maker's recursive tree construction with an ascending color pattern; Breaker's transfer of his finite hypergraph-vertex claims, including the pairing step for the star case). No parameter is fitted to any target result: the paper is a pure strategy-existence paper, so the 'fitted input called prediction' pattern does not arise. The only self-citations ([4] and [5]; [1] is a bachelor's thesis at the authors' university) are contextual: the load-bearing wish-function theorem (Theorem 3.5) is proved in full within Section 3, with the proof attributed to Arlt's thesis [1], so the citation to [4] is not needed for the derivation. No uniqueness theorem, ansatz, or rescaling is imported from the authors' prior work. The externally cited results — Beck's criterion (Theorem 4.7, [3, Theorem 2.4]), the finite Ramsey-game bound (Lemma 5.3, [7, Theorem 2.4.1]), and the bounded-degree coloring bound (Lemma 5.4, [6]) — are standard theorems with stated assumptions that do not include the paper's target results, hence constitute real independent support. A reviewer-flagged quantitative discrepancy exists in Theorem 4.9: the displayed inequality 2^(s−4)·C^(s−3)·C(s−1) < C^(s−1)/2^s − C^(s−2) is equivalent to C > (s−1)·2^(2s−4) + 2^s, which the stated bound ⌈s²·2^(s−3)⌉ does not imply (for t=0, s=4: 32 < 64). This is an arithmetically checkable proof gap that propagates to the constants in Theorems 1.4, Lemma 5.1 and the Maker half of Theorem 1.5, and it should be reported as a correctness risk under the current constants. It is not, however, a circularity: no claim in the paper is equivalent by construction to its own input or to a self-citation. The qualitative existence statements plausibly survive a constant repair (the Breaker half, Lemma 5.5, and the two-color characterization, Theorem 1.1, are unaffected). Verdict: no significant circularity; score 2 reflects only the presence of minor, non-load-bearing self-citations.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The theorems introduce several explicit constants (C, C', tau, alpha, plus the threshold in Theorem 1.5) that are chosen by hand to make the tree constructions and pigeonhole arguments work; they are listed as free parameters. The background results are standard combinatorial theorems, all cited. No new physical or mathematical entity is postulated without independent evidence.

free parameters (5)
  • C = max{ceil(s^2 * 2^{s-3}), 2^{3s+1} * t}
    Lower bound on the number of pairwise disjoint copies of S_aleph_0 in each G_i (i<=s) in Theorem 1.4. Chosen so that Beck's criterion and Lemma 4.8 apply.
  • C' = 3C*(s+1)*C^{s+1}
    Number of copies of S_aleph_0 in each G_i after augmentation; chosen large enough to guarantee room for Maker's K^s_C claim via Theorem 3.5.
  • tau = s+3t+1
    Branching bound on auxiliary trees in Theorem 4.5's recursion; must exceed the number of colors to make the wish function defined.
  • alpha = (3Cs+4)*(s+t)
    Number of auxiliary trees per hyperedge in Theorem 4.5; large enough so that a color class shrinks by the pigeonhole principle.
  • Maker threshold in Theorem 1.5 = max{2^{t*2^{(1+o_s(1))3s}}, 2^{2(1+o_s(1))s}}
    Sufficient order C_j for Maker to win; inherited from Theorem 1.4's C' requirements.
assumptions (5)
  • standard math Koenig's infinity lemma: every infinite finitely branching rooted tree contains a rooted ray.
    Invoked as Lemma 3.4 and used in Theorem 3.5, Theorem 4.5 and elsewhere to turn finite approximations into an infinite clique.
  • standard math Ramsey's theorem for edge 2-colorings of K_aleph_0.
    Used in Theorem 6.1 and in the discussion of MB_col(k,c) to assume one color class contains K_aleph_0.
  • standard math Every bounded-degree infinite graph is (Delta+1)-colorable (Brooks-type bound, Lemma 5.4).
    Used in Lemma 5.5 to define Breaker's pairing strategy on the bounded remainder G.
  • standard math Beck's hypergraph criterion (Theorem 4.7): if |E(F)| > 2^{s-3} * Delta_2(F) * |V(F)| then Maker wins MB_aux^(1)(F).
    External theorem from [3] used in Theorem 4.9 to give a sufficient condition in the finite game.
  • standard math Breaker wins the finite Ramsey game MB(n,q) when n <= (q/e) * 2^{q/2 - 1} (Lemma 5.3).
    External bound from [7] used in Lemma 5.5 to give Breaker's winning condition in Theorem 1.5.

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Pith. "Pith review of Maker-Breaker games on infinite graphs with precolored edges." pith.science (2026). https://pith.science/paper/K6OFMIDZ

@misc{pith2026260823349,
  author       = {Pith},
  title        = {Pith review of: Maker-Breaker games on infinite graphs with precolored edges},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K6OFMIDZ}},
  note         = {Machine review of arXiv:2608.23349}
}
abstract

Suppose we are given graphs $B$ and $G$. In the classical Maker-Breaker game $\text{MB}(B,G)$ two players, Maker and Breaker, alternately claim edges of $B$ and it is Maker's goal to claim a copy of $G$ in $B$, while it is Breaker's goal to prevent that. In this paper, $B$ is the countably infinite complete graph $K_{\aleph_0}$ and we are given finitely many infinite subgraphs $G_1, \dots, G_k \subseteq B$. In the color preserving game, it will be Maker's goal to claim a $K_{\aleph_0} \subseteq B$, which contains infinitely many edges of each $G_i$. We present sufficient winning conditions for both Maker and Breaker, if $k > 1$ and a full characterization of the game, if $k =1$. This partly answers a question of Bowler, Emde and Gut. In the (partially) pattern preserving game, it is Maker's goal to claim a copy $K$ of $K_{\aleph_0}$, such that $G_i \cap K$ is isomorphic to (a subgraph of) $G_i$ for all $i \in [k]$. In those games, we investigate some patterns for which Maker has a winning strategy.

Figures

Figures reproduced from arXiv: 2608.23349 by the authors.

Figure 1
Figure 1. For m = 1, if Breaker claims xz, we have to verify in the first case that Breaker has not claimed xz2 yet. If Breaker claims any edge of B not satisfying the requirements above and if there exists some fresh pair (z1, z2) ∈ P for which {x, p(z2)} is claimed by Maker or z2 is a root of some Ti with i ∈ [2ℓ], Maker claims xz2, if possible [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. One can think of G1 , . . . , Gk as graphs of different (standard) colors. For each color c, partition the copies of H in color c into t sets of equal size and recolor them with pairwise different shades of c. By definition of x, Maker can include exactly ℓ copies of H of each of the kt shades into her Kℵ0 . So she includes ℓt copies of each color in her Kℵ0 as desired. Let Gi be the disjoint union of {Gei j}j∈[t] ,… view at source ↗
Figure 3
Figure 3. For k = 2, let red and blue be color 0 and 1, respectively. Next to every vertex of T ′ n is written, which color this vertex wishes for. In the figure, we have w(r) = 1, since two of its children wish for 1. claimed all edges of K and by (iv), we have |c −1 (i) ∩ V (K)| = ℵ0. Since r ∈ V (K), K is as desired. In order to prove that Maker has claimed a copy of Kks after at most s(k+1)ks+1 moves, let b be maximal, su… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: In the figure above is depicted what Maker has constructed after the n th recursion step. Each bag T(n,e) contains a fixed amount of rooted trees, such that Maker has claimed each of their comparability graphs and connected every vertex of T(n,e) except for the roots t…
Figure 5
Figure 5. Figure 5: The figure above depicts a binary subtree of T(n,e,i) in the case of s = 3 and t = 1. Purple, blue and green vertices correspond to vertices from G1, G2 and G3, respectively and each orange triangle is a copy of T(τ, h) for pairwise different h with root rˆ 1 b ∈ G4, s…
Figure 6
Figure 6. Figure 6: T(n,e,i) : The tree given by the recursive construction. T ′ (e,i) : The maximal τ -ary rooted subtree of T(n,e,i) . T ′′ (e,i) : The maximal 4-ary rooted subtree of T ′ (e,i) , such that w(e,i) is constant on T ′′ (e,i) . T ′′′ (e,i) : The maximal binary rooted subtre…
Figure 7
Figure 7. Figure 7: The relation of Hm (e,q,ℓ,i) to e, q and ℓ. Otherwise, let r˜1, . . . , r˜δ be the roots of Hm (e ′ ,q′ ,ℓ′ ,1), . . . , Hm (e ′ ,q′ ,ℓ′ ,δ) , respectively. After Maker has connected v to Pℓ ′, there must exist a set Z ⊆ [δ] of size at least τ , such that for all z ∈ Z…
Figure 8
Figure 8. Figure 8: The red and blue edges have been claimed by Maker and Breaker, respectively. Whenever Maker tries to connect a finite star Sˆb a to HCs+b, in a first stage, Maker can connect rˆ b a only to a small subset Mb a ⊆ V (HCs+b). Let L b a ⊆ V (Sˆb a ) \ rˆ b a be the set of …
Figure 9
Figure 9. Figure 9: For uv and wx with c˜(uv) = c˜(wx) directed as above, we add the dashed crossing edges to P. contradicting (i) and if c˜(vy) = c˜(wz), we have c(vx) = c(wz), which contradicts (ii) for wx ∈ c −1 (j) with j ∈ [k]. Hence, we must have x /∈ {y, z}. u v w x = y z (a) The b…
Figure 10
Figure 10. Figure 10: The case analysis based on whether x ∈ {y, z} or x /∈ {y, z}. By symmetry of u and x (respectively y and z), we can assume that c˜(uv) = c˜(wx) (respectively c˜(vy) = c˜(wz)). Up to relabeling colors, assume that c(uv) = c(wx) = 1 and c(vy) = c(wz) = 2. By symmetry of…

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