---
_id: '14444'
abstract:
- lang: eng
text: "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."
acknowledgement: Sah and Sawhney were supported by NSF Graduate Research Fellowship
Program DGE-1745302. Sah was supported by the PD Soros Fellowship. Simkin was supported
by the Center of Mathematical Sciences and Applications at Harvard University.
article_processing_charge: Yes (in subscription journal)
article_type: original
arxiv: 1
author:
- first_name: Matthew Alan
full_name: Kwan, Matthew Alan
id: 5fca0887-a1db-11eb-95d1-ca9d5e0453b3
last_name: Kwan
orcid: 0000-0002-4003-7567
- first_name: Ashwin
full_name: Sah, Ashwin
last_name: Sah
- first_name: Mehtaab
full_name: Sawhney, Mehtaab
last_name: Sawhney
- first_name: Michael
full_name: Simkin, Michael
last_name: Simkin
citation:
ama: Kwan MA, Sah A, Sawhney M, Simkin M. Substructures in Latin squares. Israel
Journal of Mathematics. 2023;256(2):363-416. doi:10.1007/s11856-023-2513-9
apa: Kwan, M. A., Sah, A., Sawhney, M., & Simkin, M. (2023). Substructures in
Latin squares. Israel Journal of Mathematics. Springer Nature. https://doi.org/10.1007/s11856-023-2513-9
chicago: Kwan, Matthew Alan, Ashwin Sah, Mehtaab Sawhney, and Michael Simkin. “Substructures
in Latin Squares.” Israel Journal of Mathematics. Springer Nature, 2023.
https://doi.org/10.1007/s11856-023-2513-9.
ieee: M. A. Kwan, A. Sah, M. Sawhney, and M. Simkin, “Substructures in Latin squares,”
Israel Journal of Mathematics, vol. 256, no. 2. Springer Nature, pp. 363–416,
2023.
ista: Kwan MA, Sah A, Sawhney M, Simkin M. 2023. Substructures in Latin squares.
Israel Journal of Mathematics. 256(2), 363–416.
mla: Kwan, Matthew Alan, et al. “Substructures in Latin Squares.” Israel Journal
of Mathematics, vol. 256, no. 2, Springer Nature, 2023, pp. 363–416, doi:10.1007/s11856-023-2513-9.
short: M.A. Kwan, A. Sah, M. Sawhney, M. Simkin, Israel Journal of Mathematics 256
(2023) 363–416.
date_created: 2023-10-22T22:01:14Z
date_published: 2023-09-01T00:00:00Z
date_updated: 2023-10-31T11:27:30Z
day: '01'
department:
- _id: MaKw
doi: 10.1007/s11856-023-2513-9
external_id:
arxiv:
- '2202.05088'
intvolume: ' 256'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2202.05088
month: '09'
oa: 1
oa_version: Preprint
page: 363-416
publication: Israel Journal of Mathematics
publication_identifier:
eissn:
- 1565-8511
issn:
- 0021-2172
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Substructures in Latin squares
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 256
year: '2023'
...
---
_id: '14445'
abstract:
- lang: eng
text: "We prove the following quantitative Borsuk–Ulam-type result (an equivariant
analogue of Gromov’s Topological Overlap Theorem): Let X be a free ℤ/2-complex
of dimension d with coboundary expansion at least ηk in dimension 0 ≤ k < d. Then
for every equivariant map F: X →ℤ/2 ℝd, the fraction of d-simplices σ of X with
0 ∈ F (σ) is at least 2−d Π d−1k=0ηk.\r\n\r\nAs an application, we show that for
every sufficiently thick d-dimensional spherical building Y and every map f: Y
→ ℝ2d, we have f(σ) ∩ f(τ) ≠ ∅ for a constant fraction μd > 0 of pairs {σ, τ}
of d-simplices of Y. In particular, such complexes are non-embeddable into ℝ2d,
which proves a conjecture of Tancer and Vorwerk for sufficiently thick spherical
buildings.\r\n\r\nWe complement these results by upper bounds on the coboundary
expansion of two families of simplicial complexes; this indicates some limitations
to the bounds one can obtain by straighforward applications of the quantitative
Borsuk–Ulam theorem. Specifically, we prove\r\n\r\n• an upper bound of (d + 1)/2d
on the normalized (d − 1)-th coboundary expansion constant of complete (d + 1)-partite
d-dimensional complexes (under a mild divisibility assumption on the sizes of
the parts); and\r\n\r\n• an upper bound of (d + 1)/2d + ε on the normalized (d
− 1)-th coboundary expansion of the d-dimensional spherical building associated
with GLd+2(Fq) for any ε > 0 and sufficiently large q. This disproves, in a rather
strong sense, a conjecture of Lubotzky, Meshulam and Mozes."
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
- first_name: Pascal
full_name: Wild, Pascal
id: 4C20D868-F248-11E8-B48F-1D18A9856A87
last_name: Wild
citation:
ama: Wagner U, Wild P. Coboundary expansion, equivariant overlap, and crossing numbers
of simplicial complexes. Israel Journal of Mathematics. 2023;256(2):675-717.
doi:10.1007/s11856-023-2521-9
apa: Wagner, U., & Wild, P. (2023). Coboundary expansion, equivariant overlap,
and crossing numbers of simplicial complexes. Israel Journal of Mathematics.
Springer Nature. https://doi.org/10.1007/s11856-023-2521-9
chicago: Wagner, Uli, and Pascal Wild. “Coboundary Expansion, Equivariant Overlap,
and Crossing Numbers of Simplicial Complexes.” Israel Journal of Mathematics.
Springer Nature, 2023. https://doi.org/10.1007/s11856-023-2521-9.
ieee: U. Wagner and P. Wild, “Coboundary expansion, equivariant overlap, and crossing
numbers of simplicial complexes,” Israel Journal of Mathematics, vol. 256,
no. 2. Springer Nature, pp. 675–717, 2023.
ista: Wagner U, Wild P. 2023. Coboundary expansion, equivariant overlap, and crossing
numbers of simplicial complexes. Israel Journal of Mathematics. 256(2), 675–717.
mla: Wagner, Uli, and Pascal Wild. “Coboundary Expansion, Equivariant Overlap, and
Crossing Numbers of Simplicial Complexes.” Israel Journal of Mathematics,
vol. 256, no. 2, Springer Nature, 2023, pp. 675–717, doi:10.1007/s11856-023-2521-9.
short: U. Wagner, P. Wild, Israel Journal of Mathematics 256 (2023) 675–717.
date_created: 2023-10-22T22:01:14Z
date_published: 2023-09-01T00:00:00Z
date_updated: 2023-12-13T13:09:07Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1007/s11856-023-2521-9
external_id:
isi:
- '001081646400010'
file:
- access_level: open_access
checksum: fbb05619fe4b650f341cc730425dd9c3
content_type: application/pdf
creator: dernst
date_created: 2023-10-31T11:20:31Z
date_updated: 2023-10-31T11:20:31Z
file_id: '14475'
file_name: 2023_IsraelJourMath_Wagner.pdf
file_size: 623787
relation: main_file
success: 1
file_date_updated: 2023-10-31T11:20:31Z
has_accepted_license: '1'
intvolume: ' 256'
isi: 1
issue: '2'
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
page: 675-717
publication: Israel Journal of Mathematics
publication_identifier:
eissn:
- 1565-8511
issn:
- 0021-2172
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Coboundary expansion, equivariant overlap, and crossing numbers of simplicial
complexes
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 256
year: '2023'
...
---
_id: '10220'
abstract:
- lang: eng
text: "We study conditions under which a finite simplicial complex K can be mapped
to ℝd without higher-multiplicity intersections. An almost r-embedding is a map
f: K → ℝd such that the images of any r pairwise disjoint simplices of K do not
have a common point. We show that if r is not a prime power and d ≥ 2r + 1, then
there is a counterexample to the topological Tverberg conjecture, i.e., there
is an almost r-embedding of the (d +1)(r − 1)-simplex in ℝd. This improves on
previous constructions of counterexamples (for d ≥ 3r) based on a series of papers
by M. Özaydin, M. Gromov, P. Blagojević, F. Frick, G. Ziegler, and the second
and fourth present authors.\r\n\r\nThe counterexamples are obtained by proving
the following algebraic criterion in codimension 2: If r ≥ 3 and if K is a finite
2(r − 1)-complex, then there exists an almost r-embedding K → ℝ2r if and only
if there exists a general position PL map f: K → ℝ2r such that the algebraic intersection
number of the f-images of any r pairwise disjoint simplices of K is zero. This
result can be restated in terms of a cohomological obstruction and extends an
analogous codimension 3 criterion by the second and fourth authors. As another
application, we classify ornaments f: S3 ⊔ S3 ⊔ S3 → ℝ5 up to ornament concordance.\r\n\r\nIt
follows from work of M. Freedman, V. Krushkal and P. Teichner that the analogous
criterion for r = 2 is false. We prove a lemma on singular higher-dimensional
Borromean rings, yielding an elementary proof of the counterexample."
acknowledgement: Research supported by the Swiss National Science Foundation (Project
SNSF-PP00P2-138948), by the Austrian Science Fund (FWF Project P31312-N35), by the
Russian Foundation for Basic Research (Grants No. 15-01-06302 and 19-01-00169),
by a Simons-IUM Fellowship, and by the D. Zimin Dynasty Foundation Grant. We would
like to thank E. Alkin, A. Klyachko, V. Krushkal, S. Melikhov, M. Tancer, P. Teichner
and anonymous referees for helpful comments and discussions.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
- first_name: Arkadiy B.
full_name: Skopenkov, Arkadiy B.
last_name: Skopenkov
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Avvakumov S, Mabillard I, Skopenkov AB, Wagner U. Eliminating higher-multiplicity
intersections. III. Codimension 2. Israel Journal of Mathematics. 2021;245:501–534.
doi:10.1007/s11856-021-2216-z
apa: Avvakumov, S., Mabillard, I., Skopenkov, A. B., & Wagner, U. (2021). Eliminating
higher-multiplicity intersections. III. Codimension 2. Israel Journal of Mathematics.
Springer Nature. https://doi.org/10.1007/s11856-021-2216-z
chicago: Avvakumov, Sergey, Isaac Mabillard, Arkadiy B. Skopenkov, and Uli Wagner.
“Eliminating Higher-Multiplicity Intersections. III. Codimension 2.” Israel
Journal of Mathematics. Springer Nature, 2021. https://doi.org/10.1007/s11856-021-2216-z.
ieee: S. Avvakumov, I. Mabillard, A. B. Skopenkov, and U. Wagner, “Eliminating higher-multiplicity
intersections. III. Codimension 2,” Israel Journal of Mathematics, vol.
245. Springer Nature, pp. 501–534, 2021.
ista: Avvakumov S, Mabillard I, Skopenkov AB, Wagner U. 2021. Eliminating higher-multiplicity
intersections. III. Codimension 2. Israel Journal of Mathematics. 245, 501–534.
mla: Avvakumov, Sergey, et al. “Eliminating Higher-Multiplicity Intersections. III.
Codimension 2.” Israel Journal of Mathematics, vol. 245, Springer Nature,
2021, pp. 501–534, doi:10.1007/s11856-021-2216-z.
short: S. Avvakumov, I. Mabillard, A.B. Skopenkov, U. Wagner, Israel Journal of
Mathematics 245 (2021) 501–534.
date_created: 2021-11-07T23:01:24Z
date_published: 2021-10-30T00:00:00Z
date_updated: 2023-08-14T11:43:55Z
day: '30'
department:
- _id: UlWa
doi: 10.1007/s11856-021-2216-z
external_id:
arxiv:
- '1511.03501'
isi:
- '000712942100013'
intvolume: ' 245'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1511.03501
month: '10'
oa: 1
oa_version: Preprint
page: '501–534 '
project:
- _id: 26611F5C-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P31312
name: Algorithms for Embeddings and Homotopy Theory
publication: Israel Journal of Mathematics
publication_identifier:
eissn:
- 1565-8511
issn:
- 0021-2172
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '8183'
relation: earlier_version
status: public
- id: '9308'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Eliminating higher-multiplicity intersections. III. Codimension 2
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 245
year: '2021'
...
---
_id: '9578'
abstract:
- lang: eng
text: How long a monotone path can one always find in any edge-ordering of the complete
graph Kn? This appealing question was first asked by Chvátal and Komlós in 1971,
and has since attracted the attention of many researchers, inspiring a variety
of related problems. The prevailing conjecture is that one can always find a monotone
path of linear length, but until now the best known lower bound was n2/3-o(1).
In this paper we almost close this gap, proving that any edge-ordering of the
complete graph contains a monotone path of length n1-o(1).
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Matija
full_name: Bucić, Matija
last_name: Bucić
- first_name: Matthew Alan
full_name: Kwan, Matthew Alan
id: 5fca0887-a1db-11eb-95d1-ca9d5e0453b3
last_name: Kwan
orcid: 0000-0002-4003-7567
- first_name: Alexey
full_name: Pokrovskiy, Alexey
last_name: Pokrovskiy
- first_name: Benny
full_name: Sudakov, Benny
last_name: Sudakov
- first_name: Tuan
full_name: Tran, Tuan
last_name: Tran
- first_name: Adam Zsolt
full_name: Wagner, Adam Zsolt
last_name: Wagner
citation:
ama: Bucić M, Kwan MA, Pokrovskiy A, Sudakov B, Tran T, Wagner AZ. Nearly-linear
monotone paths in edge-ordered graphs. Israel Journal of Mathematics. 2020;238(2):663-685.
doi:10.1007/s11856-020-2035-7
apa: Bucić, M., Kwan, M. A., Pokrovskiy, A., Sudakov, B., Tran, T., & Wagner,
A. Z. (2020). Nearly-linear monotone paths in edge-ordered graphs. Israel Journal
of Mathematics. Springer. https://doi.org/10.1007/s11856-020-2035-7
chicago: Bucić, Matija, Matthew Alan Kwan, Alexey Pokrovskiy, Benny Sudakov, Tuan
Tran, and Adam Zsolt Wagner. “Nearly-Linear Monotone Paths in Edge-Ordered Graphs.”
Israel Journal of Mathematics. Springer, 2020. https://doi.org/10.1007/s11856-020-2035-7.
ieee: M. Bucić, M. A. Kwan, A. Pokrovskiy, B. Sudakov, T. Tran, and A. Z. Wagner,
“Nearly-linear monotone paths in edge-ordered graphs,” Israel Journal of Mathematics,
vol. 238, no. 2. Springer, pp. 663–685, 2020.
ista: Bucić M, Kwan MA, Pokrovskiy A, Sudakov B, Tran T, Wagner AZ. 2020. Nearly-linear
monotone paths in edge-ordered graphs. Israel Journal of Mathematics. 238(2),
663–685.
mla: Bucić, Matija, et al. “Nearly-Linear Monotone Paths in Edge-Ordered Graphs.”
Israel Journal of Mathematics, vol. 238, no. 2, Springer, 2020, pp. 663–85,
doi:10.1007/s11856-020-2035-7.
short: M. Bucić, M.A. Kwan, A. Pokrovskiy, B. Sudakov, T. Tran, A.Z. Wagner, Israel
Journal of Mathematics 238 (2020) 663–685.
date_created: 2021-06-21T13:24:35Z
date_published: 2020-07-01T00:00:00Z
date_updated: 2023-02-23T14:01:35Z
day: '01'
doi: 10.1007/s11856-020-2035-7
extern: '1'
external_id:
arxiv:
- '1809.01468'
intvolume: ' 238'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1809.01468
month: '07'
oa: 1
oa_version: Preprint
page: 663-685
publication: Israel Journal of Mathematics
publication_identifier:
eissn:
- 1565-8511
issn:
- 0021-2172
publication_status: published
publisher: Springer
quality_controlled: '1'
scopus_import: '1'
status: public
title: Nearly-linear monotone paths in edge-ordered graphs
type: journal_article
user_id: 6785fbc1-c503-11eb-8a32-93094b40e1cf
volume: 238
year: '2020'
...
---
_id: '9580'
abstract:
- lang: eng
text: An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H
into r parts and the size of the cut is the number of edges which have a vertex
in each part. A classical result of Edwards says that every m-edge graph has a
2-cut of size m/2+Ω)(m−−√) and this is best possible. That is, there exist cuts
which exceed the expected size of a random cut by some multiple of the standard
deviation. We study analogues of this and related results in hypergraphs. First,
we observe that similarly to graphs, every m-edge k-uniform hypergraph has an
r-cut whose size is Ω(m−−√) larger than the expected size of a random r-cut. Moreover,
in the case where k = 3 and r = 2 this bound is best possible and is attained
by Steiner triple systems. Surprisingly, for all other cases (that is, if k ≥
4 or r ≥ 3), we show that every m-edge k-uniform hypergraph has an r-cut whose
size is Ω(m5/9) larger than the expected size of a random r-cut. This is a significant
difference in behaviour, since the amount by which the size of the largest cut
exceeds the expected size of a random cut is now considerably larger than the
standard deviation.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: David
full_name: Conlon, David
last_name: Conlon
- first_name: Jacob
full_name: Fox, Jacob
last_name: Fox
- first_name: Matthew Alan
full_name: Kwan, Matthew Alan
id: 5fca0887-a1db-11eb-95d1-ca9d5e0453b3
last_name: Kwan
orcid: 0000-0002-4003-7567
- first_name: Benny
full_name: Sudakov, Benny
last_name: Sudakov
citation:
ama: Conlon D, Fox J, Kwan MA, Sudakov B. Hypergraph cuts above the average. Israel
Journal of Mathematics. 2019;233(1):67-111. doi:10.1007/s11856-019-1897-z
apa: Conlon, D., Fox, J., Kwan, M. A., & Sudakov, B. (2019). Hypergraph cuts
above the average. Israel Journal of Mathematics. Springer. https://doi.org/10.1007/s11856-019-1897-z
chicago: Conlon, David, Jacob Fox, Matthew Alan Kwan, and Benny Sudakov. “Hypergraph
Cuts above the Average.” Israel Journal of Mathematics. Springer, 2019.
https://doi.org/10.1007/s11856-019-1897-z.
ieee: D. Conlon, J. Fox, M. A. Kwan, and B. Sudakov, “Hypergraph cuts above the
average,” Israel Journal of Mathematics, vol. 233, no. 1. Springer, pp.
67–111, 2019.
ista: Conlon D, Fox J, Kwan MA, Sudakov B. 2019. Hypergraph cuts above the average.
Israel Journal of Mathematics. 233(1), 67–111.
mla: Conlon, David, et al. “Hypergraph Cuts above the Average.” Israel Journal
of Mathematics, vol. 233, no. 1, Springer, 2019, pp. 67–111, doi:10.1007/s11856-019-1897-z.
short: D. Conlon, J. Fox, M.A. Kwan, B. Sudakov, Israel Journal of Mathematics 233
(2019) 67–111.
date_created: 2021-06-21T13:36:02Z
date_published: 2019-08-01T00:00:00Z
date_updated: 2023-02-23T14:01:41Z
day: '01'
doi: 10.1007/s11856-019-1897-z
extern: '1'
external_id:
arxiv:
- '1803.08462'
intvolume: ' 233'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1803.08462
month: '08'
oa: 1
oa_version: Preprint
page: 67-111
publication: Israel Journal of Mathematics
publication_identifier:
eissn:
- 1565-8511
issn:
- 0021-2172
publication_status: published
publisher: Springer
quality_controlled: '1'
scopus_import: '1'
status: public
title: Hypergraph cuts above the average
type: journal_article
user_id: 6785fbc1-c503-11eb-8a32-93094b40e1cf
volume: 233
year: '2019'
...