TY - JOUR
T1 - Irreducible Subcube Partitions
AU - Filmus, Yuval
AU - Hirsch, Edward A.
AU - Kurz, Sascha
AU - Ihringer, Ferdinand
AU - Riazanov, Artur
AU - Smal, Alexander
AU - Vinyals, Marc
N1 - Funding Information: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 802020-ERC-HARMONIC. Ferdinand Ihringer is supported by a postdoctoral fellowship of the Research Foundation – Flanders (FWO). Edward A. Hirsch acknowledges the support of the universities he was visiting: the Technion under the above-mentioned ERC grant and the University of Haifa where his reseach was partially funded by Israel Science Foundation, ISF grant number 716/20, and funded/co-funded by the European Union (ERC, ECCC, 101076663). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. Publisher Copyright: © The authors. Released under the CC BY license (International 4.0).
PY - 2023/9/8
Y1 - 2023/9/8
N2 - A subcube partition is a partition of the Boolean cube {0, 1}n into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it “mentions” all coordinates. We study extremal properties of tight irreducible subcube partitions: minimal size, minimal weight, maximal number of points, maximal size, and maximal minimum dimension. We also consider the existence of homogeneous tight irreducible subcube partitions, in which all subcubes have the same dimensions. We additionally study subcube partitions of {0, . . . , q − 1}n, and partitions of Fn2 into affine subspaces, in both cases focusing on the minimal size. Our constructions and computer experiments lead to several conjectures on the extremal values of the aforementioned properties.
AB - A subcube partition is a partition of the Boolean cube {0, 1}n into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it “mentions” all coordinates. We study extremal properties of tight irreducible subcube partitions: minimal size, minimal weight, maximal number of points, maximal size, and maximal minimum dimension. We also consider the existence of homogeneous tight irreducible subcube partitions, in which all subcubes have the same dimensions. We additionally study subcube partitions of {0, . . . , q − 1}n, and partitions of Fn2 into affine subspaces, in both cases focusing on the minimal size. Our constructions and computer experiments lead to several conjectures on the extremal values of the aforementioned properties.
UR - https://www.scopus.com/pages/publications/85172785619
U2 - 10.37236/11862
DO - 10.37236/11862
M3 - Article
SN - 1077-8926
VL - 30
JO - Electronic Journal of Combinatorics
JF - Electronic Journal of Combinatorics
IS - 3
M1 - 3
ER -