---
res:
bibo_abstract:
- "We prove several results about substructures in Latin squares. First, we explain
how to adapt our recent work on high-girth Steiner triple systems to the setting
of Latin squares, resolving a conjecture of Linial that there exist Latin squares
with arbitrarily high girth. As a consequence, we see that the number of order-
n Latin squares with no intercalate (i.e., no 2×2 Latin subsquare) is at least
\ (e−9/4n−o(n))n2. Equivalently, P[N=0]≥e−n2/4−o(n2)=e−(1+o(1))EN\r\n , where
\ N is the number of intercalates in a uniformly random order- n Latin square.
\r\nIn fact, extending recent work of Kwan, Sah, and Sawhney, we resolve the general
large-deviation problem for intercalates in random Latin squares, up to constant
factors in the exponent: for any constant 0<δ≤1 we have P[N≤(1−δ)EN]=exp(−Θ(n2))
and for any constant δ>0 we have P[N≥(1+δ)EN]=exp(−Θ(n4/3logn)). \r\nFinally,
as an application of some new general tools for studying substructures in random
Latin squares, we show that in almost all order- n Latin squares, the number of
cuboctahedra (i.e., the number of pairs of possibly degenerate 2×2 submatrices
with the same arrangement of symbols) is of order n4, which is the minimum possible.
As observed by Gowers and Long, this number can be interpreted as measuring ``how
associative'' the quasigroup associated with the Latin square is.@eng"
bibo_authorlist:
- foaf_Person:
foaf_givenName: Matthew Alan
foaf_name: Kwan, Matthew Alan
foaf_surname: Kwan
foaf_workInfoHomepage: http://www.librecat.org/personId=5fca0887-a1db-11eb-95d1-ca9d5e0453b3
orcid: 0000-0002-4003-7567
- foaf_Person:
foaf_givenName: Ashwin
foaf_name: Sah, Ashwin
foaf_surname: Sah
- foaf_Person:
foaf_givenName: Mehtaab
foaf_name: Sawhney, Mehtaab
foaf_surname: Sawhney
- foaf_Person:
foaf_givenName: Michael
foaf_name: Simkin, Michael
foaf_surname: Simkin
bibo_doi: 10.1007/s11856-023-2513-9
bibo_issue: '2'
bibo_volume: 256
dct_date: 2023^xs_gYear
dct_isPartOf:
- http://id.crossref.org/issn/0021-2172
- http://id.crossref.org/issn/1565-8511
dct_language: eng
dct_publisher: Springer Nature@
dct_title: Substructures in Latin squares@
...