ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.07
https://doi.org/10.26493/1855-3974.2677.b7f
(Also available at http://amc-journal.eu)
The covering lemma and q-analogues of
extremal set theory problems
Dániel Gerbner *
Alfréd Rényi Institute of Mathematics, P.O.B. 127, Budapest H-1364, Hungary
Received 8 August 2021, accepted 27 January 2023, published online 22 August 2023
Abstract
We prove a general lemma (inspired by a lemma of Holroyd and Talbot) about the
connection of the largest cardinalities (or weight) of structures satisfying some hereditary
property and substructures satisfying the same hereditary property. We use it to show how
results concerning forbidden subposet problems in the Boolean poset imply analogous re-
sults in the poset of subspaces of a finite vector space. We also study generalized forbidden
subposet problems in the poset of subspaces.
Keywords: Subspace lattice, forbidden subposet, covering, profile polytope.
Math. Subj. Class. (2020): 06A07, 05D05
*Research supported by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences and the
National Research, Development and Innovation Office – NKFIH under the grants K 116769, KH 130371 and
SNN 129364.
E-mail address: gerbner.daniel@renyi.hu (Dániel Gerbner)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 1855-3966 (tiskana izd.), ISSN 1855-3974 (elektronska izd.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.07
https://doi.org/10.26493/1855-3974.2677.b7f
(Dostopno tudi na http://amc-journal.eu)
Krovna lema in q-analogije ekstremalnih
problemov v teoriji množic
Dániel Gerbner *
Alfréd Rényi Institute of Mathematics, P.O.B. 127, Budapest H-1364, Hungary
Prejeto 8. avgusta 2021, sprejeto 27. januarja 2023, objavljeno na spletu 22. avgusta 2023
Povzetek
Dokažemo splošno lemo (po vzoru leme Holroyda in Talbota) o zvezi med največjimi
kardinalnostmi (ali utežmi) struktur, ki zadoščajo nekaterim hereditarnim lastnostim, in
podstruktur z isto hereditarno lastnostjo. S pomočjo te leme pokažemo, da rezultati v
zvezi s problemi prepovedanih delno urejenih podmnožic v Booleovi delno urejeni množici
implicirajo analogne rezultate v delno urejeni množici podprostorov končnega vektorskega
prostora. Raziskujemo tudi probleme posplošenih prepovedanih delno urejenih podmnožic
v delno urejeni množici podprostorov.
Ključne besede: Mreža podprostorov, prepovedana delno urejena množica, pokritje, profilni politopi.
Math. Subj. Class. (2020): 06A07, 05D05
*Raziskavo podpira János Bolyai Research Fellowship of the Hungarian Academy of Sciences in National
Research, Development and Innovation Office – NKFIH z dotacijami K 116769, KH 130371 in SNN 129364.
E-poštni naslov: gerbner.daniel@renyi.hu (Dániel Gerbner)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/