<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this paper we define a bridge between pairings and colorings of the hypergraphs by introducing a generalization of pairs called t-cakes for t ∈ ℕ, t ≥ 2. For t = 2 the 2-cakes are the same as the well-known pairs of system of distinct representatives, that can be turned to pairing strategies in Maker-Breaker hypergraph games, see Hales and Jewett <xref ref-type="bibr" rid="j_ausm-2016-0015_ref_012_w2aab2b8c13b1b7b1ab1ac12Aa">[12]</xref>. The two-colorings are the other extremity of t-cakes, in which the whole ground set of the hypergraph is one big cake that we divide into two parts (color classes). Starting from the pairings (2-cake placement) and two-colorings we define the generalized t-cake placements where we pair p elements by q elements (p, q ∈ ℕ, 1 ≤ p, q &lt; t, p + q = t).</p><p>The method also gives bounds on the condition of winnings in certain biased Chooser-Picker games, which can be introduced similarly to Beck <xref ref-type="bibr" rid="j_ausm-2016-0015_ref_003_w2aab2b8c13b1b7b1ab1ab3Aa">[3]</xref>. We illustrate these ideas on the k-in-a-row games for different values of k on the infinite chessboard.</p></abstract>

In this paper we define a bridge between pairings and colorings of the hypergraphs by introducing a generalization of pairs called t-cakes for t ∈ ℕ, t ≥ 2. For t = 2 the 2-cakes are the same as the well-known pairs of system of distinct representatives, that can be turned to pairing strategies in Maker-Breaker hypergraph games, see Hales and Jewett [12]. The two-colorings are the other extremity of t-cakes, in which the whole ground set of the hypergraph is one big cake that we divide into two parts (color classes). Starting from the pairings (2-cake placement) and two-colorings we define the generalized t-cake placements where we pair p elements by q elements (p, q ∈ ℕ, 1 ≤ p, q < t, p + q = t).The method also gives bounds on the condition of winnings in certain biased Chooser-Picker games, which can be introduced similarly to Beck [3]. We illustrate these ideas on the k-in-a-row games for different values of k on the infinite chessboard.

Generalized pairing strategies-a bridge from pairing strategies to colorings

Department of Computer Science, University of Szeged

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Generalized pairing strategies-a bridge from pairing strategies to colorings"}

In this paper we define a bridge between pairings and colorings of the hypergraphs by introducing a generalization of pairs called t-cakes for t ∈ ℕ, t ≥ 2....

Generalized pairing strategies-a bridge from pairing strategies to colorings

Published Online: Jan 23, 2017

Page range: 233 - 248

Received: Apr 04, 2016

DOI: https://doi.org/10.1515/ausm-2016-0015

Keywords
positional games, Chooser-Picker games, k-in-a-row game, pairing strategies, hypergraph colorings

© 2016 Lajos Győrffy et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Generalized pairing strategies-a bridge from pairing strategies to colorings

Published Online: Jan 23, 2017

Page range: 233 - 248

Received: Apr 04, 2016

DOI: https://doi.org/10.1515/ausm-2016-0015

Keywordspositional games, Chooser-Picker games, k-in-a-row game, pairing strategies, hypergraph colorings

© 2016 Lajos Győrffy et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
positional games, Chooser-Picker games, k-in-a-row game, pairing strategies, hypergraph colorings