The Maker--Breaker Rado Game on a Random Set of Integers

Hancock, Robert

doi:10.1137/18m117488x

Cited by 3 publications

(4 citation statements)

References 21 publications

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“…Suppose A and B have dimensions

ℓ_{A} \times k_{A}

and

ℓ_{B} \times k_{B}

respectively. By Proposition 12 in [8], there exists vectors

a^{'}, b^{'}

such that every

k_{A}

‐distinct solution

x = (x_{1}, \dots, x_{k_{A}})

to A contains as an ordered subvector

x^{'} = (x_{i_{1}}, \dots, x_{i_{k}})

(where

i_{1} < . . . < i_{k}

), a k ‐distinct solution to

C (A) x = a^{'}

and also every

k_{B}

‐distinct solution

y = (x_{1}, \dots, x_{k_{B}})

to B contains as an ordered subvector

y^{'} = (y_{j_{1}}, \dots, y_{j_{k}})

(where

j_{1} < \dots < j_{k}

), a k ‐distinct solution to

C (B)

…”

Section: Proof Of Theorem 5 and The Ka=kb Case Of Theoremmentioning

confidence: 94%

“…Suppose A and B are irredundant matrices that satisfy ( * ). Then Proposition 4.3(iv)-(v) in [9] implies that the definitions of m(A) and m(A, B) are well-defined (i.e., have positive denominator); Proposition 12 in [8] implies that C(A) is also well-defined and satisfies ( * ) itself. Theorem 6.…”

Section: A Unifying Theoremmentioning

confidence: 99%

“…Suppose A and B have dimensions 𝓁 A × k A and 𝓁 B × k B respectively. By Proposition 12 in [8], there exists vectors a ′ , b ′ such that every k A -distinct solution x = (x 1 , … , x k A ) to A contains as an ordered subvector x ′ = (x i 1 , … , x i k ) (where i 1 < ... < i k ), a k-distinct solution to C(A)x = a ′ and also every k B -distinct solution y = (x 1 , … , x k B ) to B contains as an ordered subvector y ′ = (y j 1 , … , y j k ) (where First note that Claim 1 holds as before (with A ′ and B ′ playing the roles of A and B respectively). As in the proof of Theorem 4, we define some hypergraph notation, then prove the result by combining deterministic and probabilistic lemmas.…”

mentioning

confidence: 94%

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An asymmetric random Rado theorem for single equations: The 0‐statement

Hancock

Treglown

2021

Random Struct Algorithms

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A famous result of Rado characterizes those integer matrices A which are partition regular, that is, for which any finite coloring of the positive integers gives rise to a monochromatic solution to the equation Ax = 0. Aigner-Horev and Person recently stated a conjecture on the probability threshold for the binomial random set [n] p having the asymmetric random Rado property: given partition regular matrices A 1 , … , A r (for a fixed r ≥ 2), however one r-colors [n] p , there is always a color i ∈ [r] such that there is an i-colored solution to A i x = 0. This generalizes the symmetric case, which was resolved by Rödl and Ruciński, and Friedgut, Rödl and Schacht. Aigner-Horev and Person proved the 1-statement of their asymmetric conjecture. In this paper, we resolve the 0-statement in the case where the A i x = 0 correspond to single linear equations. Additionally we close a gap in the original proof of the 0-statement of the (symmetric) random Rado theorem.

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“…Suppose A and B have dimensions

ℓ_{A} \times k_{A}

and

ℓ_{B} \times k_{B}

respectively. By Proposition 12 in [8], there exists vectors

a^{'}, b^{'}

such that every

k_{A}

‐distinct solution

x = (x_{1}, \dots, x_{k_{A}})

to A contains as an ordered subvector

x^{'} = (x_{i_{1}}, \dots, x_{i_{k}})

(where

i_{1} < . . . < i_{k}

), a k ‐distinct solution to

C (A) x = a^{'}

and also every

k_{B}

‐distinct solution

y = (x_{1}, \dots, x_{k_{B}})

to B contains as an ordered subvector

y^{'} = (y_{j_{1}}, \dots, y_{j_{k}})

(where

j_{1} < \dots < j_{k}

), a k ‐distinct solution to

C (B)

…”

Section: Proof Of Theorem 5 and The Ka=kb Case Of Theoremmentioning

confidence: 94%

Section: A Unifying Theoremmentioning

confidence: 99%

mentioning

confidence: 94%

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An asymmetric random Rado theorem for single equations: The 0‐statement

Hancock

Treglown

2021

Random Struct Algorithms

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“…Maker wins the game if he is able to select all the vertices of one of the hyperedges from E, while Breaker wins if she is able to prevent Maker from doing so. We refer to the books of Beck [3] and of Hefetz et al [17] for more information on this game as well as to papers [15,24] for recent related developments.…”

Section: Introductionmentioning

confidence: 99%

Maker-Breaker resolving game

Kang

Klavžar

Yero

et al. 2020

Preprint

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A set of vertices W of a graph G is a resolving set if every vertex of G is uniquely determined by its vector of distances to W . In this paper, the Maker-Breaker resolving game is introduced. The game is played on a graph G by Resolver and Spoiler who alternately select a vertex of G not yet chosen. Resolver wins if at some point the vertices chosen by him form a resolving set of G, whereas Spoiler wins if the Resolver cannot form a resolving set of G. The outcome of the game is denoted by o(G) and R MB (G) (resp. S MB (G)) denotes the minimum number of moves of Resolver (resp. Spoiler) to win when Resolver has the first move. The corresponding invariants for the game when Spoiler has the first move are denoted by R ′ MB (G) and S ′ MB (G). Invariants R MB (G), R ′ MB (G), S MB (G), and S ′ MB (G) are compared among themselves and with the metric dimension dim(G). A large class of graphs G is constructed for which R MB (G) > dim(G) holds. The effect of twin equivalence classes and pairing resolving sets on the Maker-Breaker resolving game is described. As an application o(G), as well as R MB (G) and R ′ MB (G) (or S MB (G) and S ′ MB (G)), are determined for several graph classes, including trees, complete multi-partite graphs, grid graphs, and torus grid graphs.

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