ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P3.14
https://doi.org/10.26493/2590-9770.1408.f90
(Also available at http://adam-journal.eu)
Connected geometric (nk) configurations exist
for almost all n
Leah Wrenn Berman
Department of Mathematics and Statistics, University of Alaska Fairbanks, 513 Ambler
Lane, Fairbanks, AK 99775, USA
Gábor Gévay*
Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, 6720 Hungary
Tomaž Pisanski†
University of Primorska, Koper, Slovenia, and
Institute of Mathematics, Physics and Mechanics, University of Ljubljana,
Ljubljana, Slovenia
Received 10 March 2021, accepted 14 April 2021, published online 30 September 2021
Abstract
In a series of papers and in his 2009 book on configurations Branko Grünbaum de-
scribed a sequence of operations to produce new (n4) configurations from various input
configurations. These operations were later called the “Grünbaum Incidence Calculus”. We
generalize two of these operations to produce operations on arbitrary (nk) configurations.
Using them, we show that for any k there exists an integer Nk such that for any n ≥ Nk
there exists a geometric (nk) configuration. We use empirical results for k = 2, 3, 4, and
some more detailed analysis to improve the upper bound for larger values of k.
IN MEMORY OF BRANKO GRÜNBAUM
Keywords: Axial affinity, geometric configuration, Grünbaum calculus.
Math. Subj. Class.: 51A45, 51A20, 05B30, 51E30
*Corresponding author. Supported by the Hungarian National Research, Development and Innovation Office,
OTKA grant No. SNN 132625.
†Supported in part by the Slovenian Research Agency (research program P1-0294 and research projects N1-
0032, J1-9187, J1-1690, N1-0140, J1-2481), and in part by H2020 Teaming InnoRenew CoE.
E-mail addresses: lwberman@alaska.edu (Leah Wrenn Berman), gevay@math.u-szeged.hu (Gábor Gévay),
tomaz.pisanski@fmf.uni-lj.si (Tomaž Pisanski)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P3.14
https://doi.org/10.26493/2590-9770.1408.f90
(Dostopno tudi na http://adam-journal.eu)
Povezane geometrijske (nk) konfiguracije
obstajajo za skoraj vse n
Leah Wrenn Berman
Department of Mathematics and Statistics, University of Alaska Fairbanks, 513 Ambler
Lane, Fairbanks, AK 99775, USA
Gábor Gévay*
Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, 6720 Hungary
Tomaž Pisanski†
University of Primorska, Koper, Slovenia, and
Institute of Mathematics, Physics and Mechanics, University of Ljubljana,
Ljubljana, Slovenia
Prejeto 10. marca 2021, sprejeto 14. aprila 2021, objavljeno na spletu 30. septembra 2021
Povzetek
V vrsti člankov in tudi v svoji knjigi o konfiguracijah iz leta 2009 je Branko Grünbaum
opisal zaporedje operacij, s katerimi se da dobiti nove (n4) konfiguracije iz najrazličnejših
vhodnih konfiguracij. Te operacije so bile kasneje imenovane “Grünbaumov incidenčni
račun”. Posplošimo dve od teh operacij in tako dobimo operacije na poljubnih (nk) kon-
figuracijah. Z njihovo uporabo pokažemo, da za vsak k obstaja tako celo število Nk, da za
poljuben n ≥ Nk obstaja geometrijska (nk) konfiguracija. Uporabimo empirične rezultate
za k = 2, 3, 4 ter z nekaj bolj podrobnimi analizami izboljšamo zgornjo mejo za večje
vrednosti k.
V SPOMIN NA BRANKA GRÜNBAUMA
Ključne besede: Aksialna afiniteta, geometrijska konfiguracija, Grünbaumov račun.
Math. Subj. Class.: 51A45, 51A20, 05B30, 51E30
*Kontaktni avtor. Podprt s strani Hungarian National Research, Development and Innovation Office, OTKA
nepovratna sredstva št. SNN 132625.
†Delno podprt s strani Javne agencije za raziskovalno dejavnost Republike Slovenije (raziskovalni program
P1-0294 in raziskovalnih projektov N1-0032, J1-9187, J1-1690, N1-0140, J1-2481), delno pa tudi s strani H2020
Teaming InnoRenew CoE.
E-poštni naslovi: lwberman@alaska.edu (Leah Wrenn Berman), gevay@math.u-szeged.hu (Gábor Gévay),
tomaz.pisanski@fmf.uni-lj.si (Tomaž Pisanski)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/