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Fractionally Balanced Hypergraphs and Rainbow KKM Theorems

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Abstract

A d-partite hypergraph is called fractionally balanced if there exists a non-negative, not identically zero, function on its edge set that has constant degrees in each vertex side. Using a topological version of Hall’s theorem we prove lower bounds on the matching number of such hypergraphs. These bounds yield rainbow versions of the KKM theorem for products of simplices, which in turn are used to obtain some results on multiple-cake division, and on rainbow matchings in families of d-intervals.

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Abbreviations

n :

Num. of agents

d :

Num. of cakes / intervals

n t :

Num. of pieces of cake t

\({\cal P}\) :

\( = \prod _{t = 1}^d{\Delta _{{n_t} - 1}}\) set of d-cake partitions

P t :

partition of cake t; Ptnt−1

C t :

cake t / interval t (t ∈ [d])

BM :

Matching in frac. balanced hyp.

AD:

Admissible cake-division

i :

Index of an agent; i ∈ [n].

t :

Index of a cake / interval; t ∈ [d].

j t :

Piece-index in cake t; jt ∈ [nt].

\({\cal J}\) :

\( = \prod _{t = 1}^d\left[ {{n_t}} \right]\) set of piece-index vectors

P :

partition of d cakes; \(P \in {\cal P}\)

\(I_j^t\left( P \right)\) :

Interval j in partition P of cake t

IM:

Matching in interval hyp.

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Acknowledgements

We are grateful to Frédéric Meunier and Dmitry Fa-likman for helpful and instructive discussions. We also thank users Trebor, Tortar and Piquito from math stackexchange7 for their technical help. We are grateful to the anonymous referees of Combinatorica for their detailed comments on the initial version of the paper.

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Correspondence to Shira Zerbib.

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R. Aharoni was Supported by the Israel Science Foundation (ISF) grant no. 2023464 and the Discount Bank Chair at the Technion. This paper is part of a project that has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 823748. This work was supported by the Russian Federation Government in the framework of MegaGrant no. 075-15-2019-1926 when Ron Aharoni worked on Sections 6 through 8 of the paper.

Erel Segal-Halevi was supported by the Israel Science Foundation (grant no. 712/20).

Shira Zerbib was supported by NSF grant DMS-1953929.

R. Aharoni, E. Berger and S. Zerbib are supported by BSF grant no. 2016077.

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Aharoni, R., Berger, E., Briggs, J. et al. Fractionally Balanced Hypergraphs and Rainbow KKM Theorems. Combinatorica 42 (Suppl 1), 913–951 (2022). https://doi.org/10.1007/s00493-021-4808-y

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  • DOI: https://doi.org/10.1007/s00493-021-4808-y

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