Schuttes property for sets of tournaments and an application to dice games
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908950865117184 |
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| author | Jeffries, Joel |
| author_facet | Jeffries, Joel |
| contents | A tournament has Schuttes property $S_k$ if for every set of $k$ vertices, there is a vertex which dominates the set. In 1963, Erdos provided bounds for $f(k)$, the smallest order of an $S_k$ tournament. Schuttes property has various applications, including the design of unfair dice games. A set of dice introduced by James Grime motivates a generalization of Schuttes property to sets of tournaments: a set of tournaments on the same vertex set has property $S_k$ if for every set of $k$ vertices, there is a vertex which dominates the set in at least one of the tournaments. We explore this generalization and provide bounds on the fewest number of vertices needed to have an $S_k$ set of $m$ tournaments. We then apply these results to introduce a few new sets of dice similar to Grimes dice that can be used to play a game that gives one player an advantage. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_08790 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Schuttes property for sets of tournaments and an application to dice games Jeffries, Joel Combinatorics A tournament has Schuttes property $S_k$ if for every set of $k$ vertices, there is a vertex which dominates the set. In 1963, Erdos provided bounds for $f(k)$, the smallest order of an $S_k$ tournament. Schuttes property has various applications, including the design of unfair dice games. A set of dice introduced by James Grime motivates a generalization of Schuttes property to sets of tournaments: a set of tournaments on the same vertex set has property $S_k$ if for every set of $k$ vertices, there is a vertex which dominates the set in at least one of the tournaments. We explore this generalization and provide bounds on the fewest number of vertices needed to have an $S_k$ set of $m$ tournaments. We then apply these results to introduce a few new sets of dice similar to Grimes dice that can be used to play a game that gives one player an advantage. |
| title | Schuttes property for sets of tournaments and an application to dice games |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.08790 |