ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.04
https://doi.org/10.26493/1855-3974.2903.9ca
(Also available at http://amc-journal.eu)
Intersecting families of graphs of functions over
a finite field*
Angela Aguglia , Bence Csajbók †
Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari,
Via Orabona 4, I-70125 Bari, Italy
Zsuzsa Weiner
ELKH–ELTE Geometric and Algebraic Combinatorics Research Group,
1117 Budapest, Pázmány P. stny. 1/C, Hungary and
Prezi.com, H-1065 Budapest, Nagymező utca 54-56, Hungary
Received 10 June 2022, accepted 30 December 2022, published online 9 August 2023
Abstract
Let U be a set of polynomials of degree at most k over Fq , the finite field of q elements.
Assume that U is an intersecting family, that is, the graphs of any two of the polynomials in
U share a common point. Adriaensen proved that the size of U is at most qk with equality
if and only if U is the set of all polynomials of degree at most k passing through a common
point. In this manuscript, using a different, polynomial approach, we prove a stability
version of this result, that is, the same conclusion holds if |U | > qk − qk−1. We prove a
stronger result when k = 2.
For our purposes, we also prove the following results. If the set of directions determined
by the graph of f is contained in an additive subgroup of Fq , then the graph of f is a line.
If the set of directions determined by at least q −√q/2 affine points is contained in the set
of squares/non-squares plus the common point of either the vertical or the horizontal lines,
then up to an affinity the point set is contained in the graph of some polynomial of the form
αxp
k
.
Keywords: Direction problem, Erdős-Ko-Rado, finite field, polynomial.
Math. Subj. Class. (2020): 11T06
*We are extremely grateful for the reviewer’s thorough reading and valuable comments. An inaccuracy spotted
out by the reviewer led us to the discovery of Theorem 2.13. The second and the third author acknowledge the
support of the National Research, Development and Innovation Office – NKFIH, grant no. K 124950. This
work was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications
(GNSAGA– INdAM).
†Corresponding author.
E-mail addresses: angela.aguglia@poliba.it (Angela Aguglia), bence.csajbok@poliba.it (Bence Csajbók),
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) #P1.04
https://doi.org/10.26493/1855-3974.2903.9ca
(Dostopno tudi na http://amc-journal.eu)
Sekajoče se družine grafov funkcij nad končnim
obsegom*
Angela Aguglia , Bence Csajbók †
Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari,
Via Orabona 4, I-70125 Bari, Italy
Zsuzsa Weiner
ELKH–ELTE Geometric and Algebraic Combinatorics Research Group,
1117 Budapest, Pázmány P. stny. 1/C, Hungary in
Prezi.com, H-1065 Budapest, Nagymező utca 54-56, Hungary
Prejeto 10. junija 2022, sprejeto 30. decembra 2022, objavljeno na spletu 9. avgusta 2023
Povzetek
Naj bo U množica polinomov stopnje največ k nad Fq , končnim obsegom s q elementi.
Predpostavimo, da je U sekajoča se družina, kar pomeni, da imata grafa poljubnih dveh
polinomov iz U skupno točko. Adriaensen je dokazal, da je moč množice U največ qk,
kjer velja enakost natanko takrat, ko je U množica vseh polinomov stopnje največ k, ki
gredo skozi skupno točko. V tem prispevku z uporabo drugačnega, polinomskega, pristopa
dokažemo stabilnostno različico tega rezultata, kar pomeni, da isti sklep velja, če je |U | >
qk − qk−1. Dokažemo močnejši rezultat za primer, ko je k = 2.
Za naše namene dokažemo tudi naslednje rezultate. Če je množica smeri, določena z
grafom f , vsebovana v aditivni podgrupi Fq , potem je graf f premica. Če je množica smeri,
ki jo določa vsaj q −√q/2 afinih točk, vsebovana v uniji množice kvadratov/nekvadratov
in skupne točke navpičnih ali vodoravnih premic, potem je, do afinitete natančno, množica
točk vsebovana v grafu nekega polinoma oblike αxp
k
.
Ključne besede: Problem smeri, Erdős-Ko-Rado, končni obseg, polinom.
Math. Subj. Class. (2020): 11T06
zsuzsa.weiner@gmail.com (Zsuzsa Weiner)
*Zelo smo hvaležni recenzentu za temeljito branje in dragocene komentarje. Netočnost, ki jo je opazil recen-
zent, nas je pripeljala do odkritja izreka 2.13. Drugi in tretji avtor se zahvaljujeta podpori Nacionalnega urada za
raziskave, razvoj in inovacije – NKFIH, donacija št. K 124950. To delo je podprla Italijanska nacionalna skupina
za algebraične in geometrijske strukture in njihove aplikacije (GNSAGA–INdAM).
†Kontaktni avtor.
E-poštni naslovi: angela.aguglia@poliba.it (Angela Aguglia), bence.csajbok@poliba.it (Bence Csajbók),
zsuzsa.weiner@gmail.com (Zsuzsa Weiner)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/