Triangle Game in Combinatorics

Updated 10 July 2026

Triangle Game is a multifaceted concept in combinatorics, representing diverse games where triangles serve as winning targets, forbidden patterns, scoring units, or board geometries.
Research in this area employs methodologies from Maker–Breaker and avoidance games on complete graphs to bias analyses in directed tournaments and degree-preserving forcing games.
The topic offers practical insights into speed, saturation, and scoring formulations, impacting applications in network theory and combinatorial game design.

“Triangle Game” is a polysemous term in contemporary combinatorics, graph games, and recreational mathematics. In the arXiv literature it denotes several non-equivalent games whose common feature is either that triangles or directed $3$-cycles are the winning or losing configurations, or that the board itself is triangular. These include Maker–Breaker and avoidance games on $K_n$ , Waiter–Client factor games, degree-preserving Builder–Chooser forcing games, impartial token games on a directed $3$-cycle, and geometric or lattice games on triangular boards (Glazik et al., 2018, Dvořák, 2021, London, 26 May 2026, Abuku et al., 5 Sep 2025, Dickson et al., 2020).

1. Terminological scope and principal families

The term spans graph-theoretic, impartial, and geometric settings. The shared motif is not a single rule set but the role of a triangle: as a target, a forbidden pattern, a factor, a scoring unit, or the ambient geometry.

Family	Board	Objective
Maker–Breaker triangle game (Glazik et al., 2018)	$E(K_n)$	Maker claims a triangle; Breaker prevents it
Triangle avoidance / $n$ -Sim (Malekshahian, 2020)	$E(K_n)$ , two colors	First player to complete a triangle in their own color loses
Waiter–Client triangle-factor game (Dvořák, 2021)	$E(K_n)$	Waiter forces a $K_3$ -factor in Client’s graph
Degree-preserving Builder–Chooser (London, 26 May 2026)	evolving graph	Builder forces a triangle via DPG and partition steps
Constructor–Blocker triangle scoring (Boisson et al., 7 Oct 2025)	$E(K_n)$	Constructor maximizes triangles under $F$ -free or planar constraints
Digraph Triangular Nim / “Triangle Game” (Abuku et al., 5 Sep 2025)	directed $K_n$ 0-cycle with tokens	Normal- and misère-play impartial game
Triangle-board and lattice games (0711.0486, Dickson et al., 2020, Hong et al., 2023)	triangular boards or tilings	Sweeps, area claims, or Life dynamics on triangle-based boards

This dispersion of meanings is structurally significant. In some papers, “triangle game” means an achievement game on edges of a complete graph; in others it denotes an avoidance game, a speed-of-forcing problem, or a triangle-board puzzle. A precise identification of board, move rule, and win condition is therefore essential.

2. Positional graph games with triangles as target or hazard

In the classical Maker–Breaker $K_n$ 1-triangle game, Maker claims one edge per round, Breaker claims $K_n$ 2, and Maker wins by occupying all three edges of a triangle in $K_n$ 3. Chvátal and Erdős proved that Maker wins for $K_n$ 4, while Breaker wins for $K_n$ 5. Balogh and Samotij later improved Breaker’s constant to $K_n$ 6, and a deterministic potential-based strategy reduced it further: for sufficiently large $K_n$ 7, Breaker wins whenever $K_n$ 8 (Glazik et al., 2018). The exact leading constant in the threshold bias remains unknown; the paper explicitly notes that it is not known for any graph with a cycle, and that even existence of a limiting constant is open (Glazik et al., 2018).

A directed analogue replaces $K_n$ 9 by a tournament and counts only directed triangles, i.e. directed $3$0-cycles, as winning sets. On the parity tournament $3$1, defined by orienting $3$2 as $3$3 iff $3$4 is odd, the unbiased threshold is exact: Breaker wins for $3$5, whereas Maker wins for $3$6. For the $3$7 biased game on $3$8, the bias threshold satisfies

$3$9

The same paper proves that for any fixed $E(K_n)$ 0, Maker wins on the random tournament $E(K_n)$ 1 with probability tending to $E(K_n)$ 2, and introduces a flip-biased model in which Breaker first flips $E(K_n)$ 3 edge directions; the flip-bias threshold satisfies $E(K_n)$ 4 (Jagtap et al., 15 Oct 2025).

A misère triangle game appears as $E(K_n)$ 5-Sim. Two players alternately color previously uncolored edges of $E(K_n)$ 6, and the first player to create a monochromatic triangle in their own color loses. For $E(K_n)$ 7, Ramsey’s theorem implies that the game cannot end in a draw, whereas $E(K_n)$ 8-Sim has a unique draw position up to isomorphism, the “drawn $E(K_n)$ 9” (Malekshahian, 2020). Several mid-game positions admit second-player wins by strategy stealing: a decomposition into disjoint drawn $n$ 0’s when $n$ 1 is a multiple of $n$ 2, a drawn $n$ 3 minus one red edge, and a specific three-move position $n$ 4. By contrast, a drawn $n$ 5 plus two isolated vertices is a first-player win in $n$ 6-Sim (Malekshahian, 2020). The notable methodological point is that the paper adapts strategy stealing to a misère setting by “ignoring” one dangerous edge and “pretending” another exists, with explicit “insurance” that the discrepancy will never be exploited.

Online Ramsey variants again reconfigure the triangle objective. In the Builder–Painter online Ramsey game for $n$ 7 on $n$ 8-free graphs, Builder adds an edge provided the evolving graph remains $n$ 9-free, and Painter colors it red or blue; Builder wants to force a monochromatic triangle. The paper classifies all graphs $E(K_n)$ 0 with no isolated vertices, except one specific graph $E(K_n)$ 1: Painter wins on $E(K_n)$ 2-free graphs iff $E(K_n)$ 3 is isomorphic to a subgraph of one of $E(K_n)$ 4, and Painter also wins on $E(K_n)$ 5-minor-free graphs (Choi et al., 2019). This makes the triangle game a structural classification problem on graph classes rather than merely a threshold question on $E(K_n)$ 6.

3. Speed, saturation, and scoring formulations

A distinct line of work asks not whether a triangle-based condition is eventually forced, but how quickly or how profitably it can be forced.

In the unbiased Waiter–Client triangle-factor game on $E(K_n)$ 7, Waiter offers two unclaimed edges each round, Client chooses one for his graph, and Waiter gets the other. Waiter wins if Client’s graph ever contains a $E(K_n)$ 8-factor. If both players are optimal and Waiter wants to win as fast as possible while Client delays, the optimal duration is denoted $E(K_n)$ 9. Clemens et al. had proved

$E(K_n)$ 0

and conjectured equality in the asymptotic sense. The note (Dvořák, 2021) verifies the conjecture by proving the lower bound $E(K_n)$ 1, hence

$E(K_n)$ 2

The proof is component-theoretic: connected components of Client’s graph are classified as “good” or “bad”, crucial edges are tracked, and a counting argument shows that at most $E(K_n)$ 3 good components can ever be created under Client’s delaying strategy (Dvořák, 2021).

In Hajnal’s triangle-free saturation game, players alternately add edges to the empty graph on $E(K_n)$ 4 vertices, with the sole legality condition that no move may create a triangle; the game ends at a maximal triangle-free graph. One player wants the terminal graph to be as sparse as possible. The best published upper bound is

$E(K_n)$ 5

proved by forcing at least $E(K_n)$ 6 disjoint copies of $E(K_n)$ 7 and then exploiting the fact that triangle-free graphs admit at most $E(K_n)$ 8 edges between two disjoint $E(K_n)$ 9’s and at most $K_3$ 0 edges between a vertex and a $K_3$ 1 (Biró et al., 2014). This is a density-control triangle game rather than a triangle-creation game.

Constructor–Blocker games turn triangles into the score itself: Constructor claims edges first, must keep her own graph $K_3$ 2-free (or planar), and aims to maximize the number of triangles in her graph at the end, while Blocker minimizes. The resulting values $K_3$ 3 are asymptotically sharp in several regimes (Boisson et al., 7 Oct 2025).

Constraint on Constructor	Result for $K_3$ 4	Static benchmark
$K_3$ 5	$K_3$ 6	$K_3$ 7
$K_3$ 8-free	$K_3$ 9	$E(K_n)$ 0
$E(K_n)$ 1-free	$E(K_n)$ 2	$E(K_n)$ 3
$E(K_n)$ 4-free	$E(K_n)$ 5	$E(K_n)$ 6
$E(K_n)$ 7-free	between $E(K_n)$ 8 and $E(K_n)$ 9	$F$ 0
PCB	$F$ 1	planar maximum $F$ 2
ECB	between $F$ 3 and $F$ 4	outerplanar maximum $F$ 5

The comparative pattern is instructive. In some constrained triangle games, Blocker collapses a linear static extremal value to $F$ 6 ( $F$ 7, $F$ 8). In others, Constructor still achieves a positive fraction of the generalized Turán maximum. In the planar PCB variant, Constructor asymptotically matches the planar extremal triangle count by iterated wheel and fan constructions (Boisson et al., 7 Oct 2025).

4. Local forcing and one-round triangle criteria

The degree-preserving Builder–Chooser game replaces edge-claiming on a fixed graph by controlled local rewiring. Starting from a graph $F$ 9, Builder chooses a matching $K_n$ 00, deletes it, introduces a new vertex $K_n$ 01, joins $K_n$ 02 to every endpoint of $K_n$ 03, partitions the entire new edge set into two parts, and Chooser keeps one part. The forcing time $K_n$ 04 is the least number of rounds in which Builder can guarantee a copy of $K_n$ 05 regardless of Chooser’s responses (London, 26 May 2026).

For triangles on triangle-free seeds, the central parameter is

$K_n$ 06

where $K_n$ 07 is the matching number. The one-round criterion is exact: $K_n$ 08 for triangle-free $K_n$ 09 (London, 26 May 2026). The interpretation is that Builder can force a triangle in one round exactly when some matching $K_n$ 10 leaves two disjoint “cross edges” in the endpoint-induced graph; after the DPG step these become two edge-disjoint triangles through the new vertex.