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An example of a hexagon cut into three pieces of smaller diameter.

The Borsuk problem in geometry, for historical reasons^{[note 1]} incorrectly called Borsuk's conjecture, is a question in discrete geometry. It is named after Karol Borsuk.

Problem

In 1932, Karol Borsuk showed^[2] that an ordinary 3-dimensional ball in Euclidean space can be easily dissected into 4 solids, each of which has a smaller diameter than the ball, and generally $n$ -dimensional ball can be covered with $n + 1$ compact sets of diameters smaller than the ball. At the same time he proved that $n$ subsets are not enough in general. The proof is based on the Borsuk–Ulam theorem. That led Borsuk to a general question:^[2]

Die folgende Frage bleibt offen: Lässt sich jede beschränkte Teilmenge $E$ des Raumes $\mathbb {R} ^{n}$ in ( $n + 1$ ) Mengen zerlegen, von denen jede einen kleineren Durchmesser als $E$ hat?
The following question remains open: Can every bounded subset $E$ of the space $\mathbb {R} ^{n}$ be partitioned into ( $n + 1$ ) sets, each of which has a smaller diameter than $E$ ?

— Drei Sätze über die n-dimensionale euklidische Sphäre

The question was answered in the positive in the following cases:

$n = 2$ — which is the original result by Karol Borsuk (1932).
$n = 3$ — shown by Julian Perkal (1947),^[3] and independently, 8 years later, by H. G. Eggleston (1955).^[4] A simple proof was found later by Branko Grünbaum and Aladár Heppes.
For all $n$ for smooth convex fields — shown by Hugo Hadwiger (1946).^[5]^[6]
For all $n$ for centrally-symmetric fields — shown by A.S. Riesling (1971).^[7]
For all $n$ for fields of revolution — shown by Boris Dekster (1995).^[8]

The problem was finally solved in 1993 by Jeff Kahn and Gil Kalai, who showed that the general answer to Borsuk's question is no.^[9] They claim that their construction shows that $n + 1$ pieces do not suffice for $n = 1325$ and for each $n > 2014$ . However, as pointed out by Bernulf Weißbach,^[10] the first part of this claim is in fact false. But after improving a suboptimal conclusion within the corresponding derivation, one can indeed verify one of the constructed point sets as a counterexample for $n = 1325$ (as well as all higher dimensions up to 1560).^[11]

Their result was improved in 2003 by Hinrichs and Richter, who constructed finite sets for $n \geq 298$ , which cannot be partitioned into $n + 11$ parts of smaller diameter.^[1]

In 2013, Andriy V. Bondarenko had shown that Borsuk's conjecture is false for all $n \geq 65$ .^[12] Shortly after, Thomas Jenrich derived a 64-dimensional counterexample from Bondarenko's construction, giving the best bound up to now.^[13]^[14]

Apart from finding the minimum number $n$ of dimensions such that the number of pieces $α (n) > n + 1$ , mathematicians are interested in finding the general behavior of the function $α (n)$ . Kahn and Kalai show that in general (that is, for $n$ sufficiently large), one needs ${\textstyle \alpha (n)\geq (1.2)^{\sqrt {n}}}$ many pieces. They also quote the upper bound by Oded Schramm, who showed that for every $ε$ , if $n$ is sufficiently large, ${\textstyle \alpha (n)\leq \left({\sqrt {3/2}}+\varepsilon \right)^{n}}$ .^[15] The correct order of magnitude of $α (n)$ is still unknown.^[16] However, it is conjectured that there is a constant $c > 1$ such that $α (n) > c n$ for all $n \geq 1$ .

Note

^ As Hinrichs and Richter say in the introduction to their work,^[1] the "Borsuk's conjecture [was] believed by many to be true for some decades" (hence commonly called a conjecture) so "it came as a surprise when Kahn and Kalai constructed finite sets showing the contrary". It's worth noting, however, that Karol Borsuk has formulated the problem just as a question, not suggesting the expected answer would be positive.

References

^ ^a ^b Hinrichs, Aicke; Richter, Christian (28 August 2003). "New sets with large Borsuk numbers". Discrete Mathematics. 270 (1–3). Elsevier: 137–147. doi:10.1016/S0012-365X(02)00833-6.
^ ^a ^b Borsuk, Karol (1933), "Drei Sätze über die n-dimensionale euklidische Sphäre" [Three theorems about the n-dimensional Euclidean sphere] (PDF), Fundamenta Mathematicae (in German), 20: 177–190, doi:10.4064/fm-20-1-177-190
^ Perkal, Julian (1947), "Sur la subdivision des ensembles en parties de diamètre inférieur", Colloquium Mathematicum (in French), 2: 45
^ Eggleston, H. G. (1955), "Covering a three-dimensional set with sets of smaller diameter", Journal of the London Mathematical Society, 30: 11–24, doi:10.1112/jlms/s1-30.1.11, MR 0067473
^ Hadwiger, Hugo (1945), "Überdeckung einer Menge durch Mengen kleineren Durchmessers", Commentarii Mathematici Helvetici (in German), 18 (1): 73–75, doi:10.1007/BF02568103, MR 0013901, S2CID 122199549
^ Hadwiger, Hugo (1946), "Mitteilung betreffend meine Note: Überdeckung einer Menge durch Mengen kleineren Durchmessers", Commentarii Mathematici Helvetici (in German), 19 (1): 72–73, doi:10.1007/BF02565947, MR 0017515, S2CID 121053805
^ Riesling, A. S. (1971), "Проблема Борсука в трехмерных пространствах постоянной кривизны" [Borsuk's problem in three-dimensional spaces of constant curvature] (PDF), Ukr. Geom. Sbornik (in Russian), 11, Kharkov State University (now National University of Kharkiv): 78–83
^ Dekster, Boris (1995), "The Borsuk conjecture holds for fields of revolution", Journal of Geometry, 52 (1–2): 64–73, doi:10.1007/BF01406827, MR 1317256, S2CID 121586146
^ Kahn, Jeff; Kalai, Gil (1993), "A counterexample to Borsuk's conjecture", Bulletin of the American Mathematical Society, 29 (1): 60–62, arXiv:math/9307229, doi:10.1090/S0273-0979-1993-00398-7, MR 1193538, S2CID 119647518
^ Weißbach, Bernulf (2000), "Sets with Large Borsuk Number" (PDF), Beiträge zur Algebra und Geometrie (in German), 41 (2): 417–423
^ Jenrich, Thomas (2018), On the counterexamples to Borsuk's conjecture by Kahn and Kalai, arXiv:1809.09612v4
^ Bondarenko, Andriy (2014) [2013], "On Borsuk's Conjecture for Two-Distance Sets", Discrete & Computational Geometry, 51 (3): 509–515, arXiv:1305.2584, doi:10.1007/s00454-014-9579-4, MR 3201240
^ Jenrich, Thomas (2013), A 64-dimensional two-distance counterexample to Borsuk's conjecture, arXiv:1308.0206, Bibcode:2013arXiv1308.0206J
^ Jenrich, Thomas; Brouwer, Andries E. (2014), "A 64-Dimensional Counterexample to Borsuk's Conjecture", Electronic Journal of Combinatorics, 21 (4): #P4.29, doi:10.37236/4069, MR 3292266
^ Schramm, Oded (1988), "Illuminating sets of constant width", Mathematika, 35 (2): 180–189, doi:10.1112/S0025579300015175, MR 0986627
^ Alon, Noga (2002), "Discrete mathematics: methods and challenges", Proceedings of the International Congress of Mathematicians, Beijing, 1: 119–135, arXiv:math/0212390, Bibcode:2002math.....12390A

External links

Weisstein, Eric W. "Borsuk's Conjecture". MathWorld.

[WhyConjecture-2] As Hinrichs and Richter say in the introduction to their work,^[1] the "Borsuk's conjecture [was] believed by many to be true for some decades" (hence commonly called a conjecture) so "it came as a surprise when Kahn and Kalai constructed finite sets showing the contrary". It's worth noting, however, that Karol Borsuk has formulated the problem just as a question, not suggesting the expected answer would be positive.

[HinrRicht-1] Hinrichs, Aicke; Richter, Christian (28 August 2003). "New sets with large Borsuk numbers". Discrete Mathematics. 270 (1–3). Elsevier: 137–147. doi:10.1016/S0012-365X(02)00833-6.

[BorsukFM-3] Borsuk, Karol (1933), "Drei Sätze über die n-dimensionale euklidische Sphäre" [Three theorems about the n-dimensional Euclidean sphere] (PDF), Fundamenta Mathematicae (in German), 20: 177–190, doi:10.4064/fm-20-1-177-190

[4] Perkal, Julian (1947), "Sur la subdivision des ensembles en parties de diamètre inférieur", Colloquium Mathematicum (in French), 2: 45

[5] Eggleston, H. G. (1955), "Covering a three-dimensional set with sets of smaller diameter", Journal of the London Mathematical Society, 30: 11–24, doi:10.1112/jlms/s1-30.1.11, MR 0067473

[6] Hadwiger, Hugo (1945), "Überdeckung einer Menge durch Mengen kleineren Durchmessers", Commentarii Mathematici Helvetici (in German), 18 (1): 73–75, doi:10.1007/BF02568103, MR 0013901, S2CID 122199549

[7] Hadwiger, Hugo (1946), "Mitteilung betreffend meine Note: Überdeckung einer Menge durch Mengen kleineren Durchmessers", Commentarii Mathematici Helvetici (in German), 19 (1): 72–73, doi:10.1007/BF02565947, MR 0017515, S2CID 121053805

[8] Riesling, A. S. (1971), "Проблема Борсука в трехмерных пространствах постоянной кривизны" [Borsuk's problem in three-dimensional spaces of constant curvature] (PDF), Ukr. Geom. Sbornik (in Russian), 11, Kharkov State University (now National University of Kharkiv): 78–83

[9] Dekster, Boris (1995), "The Borsuk conjecture holds for fields of revolution", Journal of Geometry, 52 (1–2): 64–73, doi:10.1007/BF01406827, MR 1317256, S2CID 121586146

[10] Kahn, Jeff; Kalai, Gil (1993), "A counterexample to Borsuk's conjecture", Bulletin of the American Mathematical Society, 29 (1): 60–62, arXiv:math/9307229, doi:10.1090/S0273-0979-1993-00398-7, MR 1193538, S2CID 119647518

[11] Weißbach, Bernulf (2000), "Sets with Large Borsuk Number" (PDF), Beiträge zur Algebra und Geometrie (in German), 41 (2): 417–423

[12] Jenrich, Thomas (2018), On the counterexamples to Borsuk's conjecture by Kahn and Kalai, arXiv:1809.09612v4

[13] Bondarenko, Andriy (2014) [2013], "On Borsuk's Conjecture for Two-Distance Sets", Discrete & Computational Geometry, 51 (3): 509–515, arXiv:1305.2584, doi:10.1007/s00454-014-9579-4, MR 3201240

[14] Jenrich, Thomas (2013), A 64-dimensional two-distance counterexample to Borsuk's conjecture, arXiv:1308.0206, Bibcode:2013arXiv1308.0206J

[15] Jenrich, Thomas; Brouwer, Andries E. (2014), "A 64-Dimensional Counterexample to Borsuk's Conjecture", Electronic Journal of Combinatorics, 21 (4): #P4.29, doi:10.37236/4069, MR 3292266

[16] Schramm, Oded (1988), "Illuminating sets of constant width", Mathematika, 35 (2): 180–189, doi:10.1112/S0025579300015175, MR 0986627

[17] Alon, Noga (2002), "Discrete mathematics: methods and challenges", Proceedings of the International Congress of Mathematicians, Beijing, 1: 119–135, arXiv:math/0212390, Bibcode:2002math.....12390A

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@@ Line 1: / Line 1: @@
+{{short description|Can every bounded subset of Rn be partitioned into (n+1) smaller diameter sets?}}
 [[File:Borsuk Hexagon.svg|200px|thumb|right|An example of a [[hexagon]] cut into three pieces of  smaller diameter.]]
-The '''Borsuk problem in geometry''', for historical reasons{{refn|group=note|name="WhyConjecture"|As Hinrichs and Richter say in the introduction to their work<ref name=HinrRicht />, the ''“Borsuk's conjecture [was] believed by many to be true for some decades”'' and hence commonly called 'a conjecture'. It's worth noting, however, that Karol Borsuk has formulated the problem just as a question, not leaning either to 'yes' or 'no'.}} incorrectly called '''Borsuk's [[conjecture]]''', is a question in [[discrete geometry]].
+The '''Borsuk problem in geometry''', for historical reasons{{refn|group=note|name="WhyConjecture"|As Hinrichs and Richter say in the introduction to their work,<ref name=HinrRicht /> the "Borsuk's conjecture [was] believed by many to be true for some decades" (hence commonly called a ''conjecture'') so "it came as a surprise when Kahn and Kalai constructed finite sets showing the contrary". It's worth noting, however, that Karol Borsuk has formulated the problem just as a question, not suggesting the expected answer would be positive.}} incorrectly called '''Borsuk's [[conjecture]]''', is a question in [[discrete geometry]]. It is named after [[Karol Borsuk]].
 ==Problem==
-In 1932 [[Karol Borsuk]] showed<ref name="BorsukFM">{{citation
+In 1932, [[Karol Borsuk]] showed<ref name="BorsukFM">{{citation
  | last = Borsuk | first = Karol
  | journal = [[Fundamenta Mathematicae]] | authorlink = Karol Borsuk
  | pages = 177–190
  | title = Drei Sätze über die n-dimensionale euklidische Sphäre
+ | trans-title = Three theorems about the n-dimensional Euclidean sphere
  | language = de
  | volume = 20
  | year = 1933
- | url = http://matwbn.icm.edu.pl/ksiazki/fm/fm20/fm20117.pdf}}</ref> that an ordinary 3-dimensional [[ball (mathematics)|ball]] in [[Euclidean space]] can be easily dissected into 4 solids, each of which has a smaller [[diameter]] than the ball, and generally ''d''-dimensional ball can be covered with {{nobr|''d'' + 1}} [[Compact space|compact]] [[Set (mathematics)|sets]] of diameters smaller than the ball. At the same time he proved that ''d'' [[subset]]s are not enough in general. The proof is based on the [[Borsuk–Ulam theorem]]. That led Borsuk to a general question:
+ | url = http://matwbn.icm.edu.pl/ksiazki/fm/fm20/fm20117.pdf| doi = 10.4064/fm-20-1-177-190
+ | doi-access = free
+ }}</ref> that an ordinary 3-dimensional [[ball (mathematics)|ball]] in [[Euclidean space]] can be easily dissected into 4 solids, each of which has a smaller [[diameter]] than the ball, and generally {{mvar|n}}-dimensional ball can be covered with {{math|''n'' + 1}} [[Compact space|compact]] [[Set (mathematics)|sets]] of diameters smaller than the ball. At the same time he proved that {{mvar|n}} [[subset]]s are not enough in general. The proof is based on the [[Borsuk–Ulam theorem]]. That led Borsuk to a general question:<ref name="BorsukFM" />
+{{blockquote |text=
-: ''Die folgende Frage bleibt offen: Lässt sich jede beschränkte Teilmenge E des Raumes <math>\Bbb R^n</math> in (n&nbsp;+&nbsp;1) Mengen zerlegen, von denen jede einen kleineren Durchmesser als E hat?''<ref name="BorsukFM" />
+''{{lang|de|text= Die folgende Frage bleibt offen: Lässt sich jede beschränkte Teilmenge {{mvar|E}} des Raumes <math>\mathbb{R}^n</math> in ({{math|n + 1}}) Mengen zerlegen, von denen jede einen kleineren Durchmesser als {{mvar|E}} hat? |italic= unset}}''
+The following question remains open: Can every [[bounded set|bounded]] subset {{mvar|E}} of the space <math>\mathbb{R}^n</math> be [[partition of a set|partitioned]] into ({{math|''n'' + 1}}) sets, each of which has a smaller diameter than {{mvar|E}}?
-This can be translated as:
+|multiline= yes
+|source= {{lang|de|Drei Sätze über die n-dimensionale euklidische Sphäre}}
-: ''The following question remains open: Can every [[bounded set|bounded]] subset E of the space <math>\Bbb R^n</math> be [[partition of a set|partitioned]] into (n&nbsp;+&nbsp;1) sets, each of which has a smaller diameter than E?''
+}}
-The question got a positive answer in the following cases:
+The question was answered in the positive in the following cases:
-* ''d'' = 2 — which is the original result by Karol Borsuk (1932).
+* {{math|1= ''n'' = 2}} — which is the original result by Karol Borsuk (1932).
-* ''d'' = 3 — shown by Julian Perkal (1947),<ref>{{citation
+* {{math|1= ''n'' = 3}} — shown by Julian Perkal (1947),<ref>{{citation
  | last = Perkal | first = Julian
  | journal = Colloquium Mathematicum
@@ Line 28: / Line 34: @@
  | title = Sur la subdivision des ensembles en parties de diamètre inférieur
  | volume = 2
- | year = 1947}}</ref> and independently, 8 years later, by H. G. Eggleston (1955).<ref>{{citation
+ | year = 1947
+ | lang = fr}}</ref> and independently, 8 years later, by H. G. Eggleston (1955).<ref>{{citation
  | last = Eggleston | first = H. G.
  | mr = 0067473
@@ Line 37: / Line 44: @@
  | year = 1955
  | doi = 10.1112/jlms/s1-30.1.11}}</ref> A simple proof was found later by [[Branko Grünbaum]] and Aladár Heppes.
-* For all ''d'' for the [[Smooth manifold|smooth]] convex bodies — shown by [[Hugo Hadwiger]] (1946).<ref>{{citation
+* For all {{mvar|n}} for [[Smooth manifold|smooth]] convex fields — shown by [[Hugo Hadwiger]] (1946).<ref>{{citation
  | last = Hadwiger | first = Hugo | authorlink = Hugo Hadwiger
  | mr = 0013901
@@ Line 46: / Line 53: @@
  | issue = 1
  | year = 1945
- | doi = 10.1007/BF02568103}}</ref><ref>{{citation
+ | lang = de
+ | doi = 10.1007/BF02568103| s2cid = 122199549 }}</ref><ref>{{citation
  | last = Hadwiger | first = Hugo | authorlink = Hugo Hadwiger
  | mr = 0017515
@@ Line 55: / Line 63: @@
  | issue = 1
  | year = 1946
+ | lang = de
- | doi = 10.1007/BF02565947}}</ref>
+ | doi = 10.1007/BF02565947| s2cid = 121053805 }}</ref>
-* For all ''d'' for [[Central symmetry|centrally-symmetric]] bodies — shown by A.S. Riesling (1971).<ref>{{citation
+* For all {{mvar|n}} for [[Central symmetry|centrally-symmetric]] fields — shown by A.S. Riesling (1971).<ref>{{citation
  | url = http://geometry.karazin.ua/resources/articles/594b744b34d8b035cdea7128bbae7d64.pdf
  | language = ru
@@ Line 63: / Line 72: @@
  | publisher = Kharkov State University (now [[National University of Kharkiv]])
  | pages = 78–83
+ | title = Проблема Борсука в трехмерных пространствах постоянной кривизны
- | title = Borsuk's problem in three-dimensional spaces of constant curvature
+ | trans-title = Borsuk's problem in three-dimensional spaces of constant curvature
  | volume = 11
  | year = 1971}}</ref>
-* For all ''d'' for [[Solid of revolution|bodies of revolution]] — shown by Boris Dekster (1995).<ref>{{citation
+* For all {{mvar|n}} for [[Solid of revolution|fields of revolution]] — shown by Boris Dekster (1995).<ref>{{citation
  | last = Dekster | first = Boris
  | mr = 1317256
  | journal = Journal of Geometry
  | pages = 64–73
- | title = The Borsuk conjecture holds for bodies of revolution
+ | title = The Borsuk conjecture holds for fields of revolution
  | volume = 52
- | issue = 1-2
+ | issue = 1–2
  | year = 1995
- | doi = 10.1007/BF01406827}}</ref>
+ | doi = 10.1007/BF01406827| s2cid = 121586146
+ }}</ref>
-The problem was finally solved in 1993 by [[Jeff Kahn]] and [[Gil Kalai]], who showed that the general answer to Borsuk's question is ''no''.<ref>{{citation
+The problem was finally solved in 1993 by [[Jeff Kahn]] and [[Gil Kalai]], who showed that the general answer to Borsuk's question is {{em|no}}.<ref>{{citation
  | last1 = Kahn | first1 = Jeff | authorlink1 = Jeff Kahn
  | last2 = Kalai | first2 = Gil | authorlink2 = Gil Kalai
@@ Line 88: / Line 99: @@
  | year = 1993
  | arxiv = math/9307229
- | doi = 10.1090/S0273-0979-1993-00398-7}}</ref> Their construction shows that {{nobr|''d'' + 1}} pieces do not suffice for {{nobr|1=''d'' = 1,325}} and for each {{nobr|''d'' > 2,014}}.
+ | doi = 10.1090/S0273-0979-1993-00398-7| s2cid = 119647518 }}</ref> They claim that their construction shows that {{math|''n'' + 1}} pieces do not suffice for {{math|1=''n'' = 1325}} and for each {{math|''n'' > 2014}}. However, as pointed out by Bernulf Weißbach,<ref>{{citation
+ | url = https://www.emis.de/journals/BAG/vol.41/no.2/b41h2wb1.pdf
+ | last1 = Weißbach | first1 = Bernulf
+ | journal = Beiträge zur Algebra und Geometrie
+ | pages = 417–423
+ | title = Sets with Large Borsuk Number
+ | volume = 41
+ | issue = 2
+ | year = 2000
+ | lang = de}}</ref> the first part of this claim is in fact false. But after improving a suboptimal conclusion within the corresponding derivation, one can indeed verify one of the constructed point sets as a counterexample for {{math|1=''n'' = 1325}} (as well as all higher dimensions up to 1560).<ref>{{citation
+ | last1 = Jenrich | first1 = Thomas
+ | title = On the counterexamples to Borsuk's conjecture by Kahn and Kalai
+ | year = 2018
+ | arxiv = 1809.09612v4}}</ref>
+Their result was improved in 2003 by Hinrichs and Richter, who constructed finite sets for {{math|''n'' ≥ 298}}, which cannot be partitioned into {{math|''n'' + 11}} parts of smaller diameter.<ref name=HinrRicht />
-After Andriy V. Bondarenko had shown that Borsuk’s conjecture is false for all {{nobr|''d'' ≥ 65}},<ref>{{citation
- | last = Bondarenko | first = Andriy V.
+In 2013, Andriy V. Bondarenko had shown that Borsuk's conjecture is false for all {{math|''n'' ≥ 65}}.<ref>
- | title = On Borsuk’s conjecture for two-distance sets
+{{citation
- | year = 2013
- | arxiv = 1305.2584}}</ref>
-<ref>{{citation
  | last = Bondarenko | first = Andriy
  | mr = 3201240
  | journal = [[Discrete & Computational Geometry]]
  | pages = 509–515
- | title = On Borsuk’s Conjecture for Two-Distance Sets
+ | title = On Borsuk's Conjecture for Two-Distance Sets
  | volume = 51
  | issue = 3
- | year = 2014
+ | orig-date = 2013
+ | date = 2014
- | doi = 10.1007/s00454-014-9579-4}}</ref> the current best bound, due to Thomas Jenrich, is 64.<ref>{{citation
+ | doi = 10.1007/s00454-014-9579-4
+ | doi-access = free
+ | arxiv = 1305.2584}}</ref> Shortly after, Thomas Jenrich derived a 64-dimensional counterexample from Bondarenko's construction, giving the best bound up to now.<ref>{{citation
  | last = Jenrich | first = Thomas
  | title = A 64-dimensional two-distance counterexample to Borsuk's conjecture
  | year = 2013
- | arxiv = 1308.0206}}</ref><ref>{{citation
+ | arxiv = 1308.0206| bibcode = 2013arXiv1308.0206J}}</ref><ref>{{citation
  | last1 = Jenrich | first1 = Thomas
  | last2 = Brouwer | first2 = Andries E. | authorlink2 = Andries Brouwer
@@ Line 117: / Line 142: @@
  | volume = 21
  | issue = 4
- | year = 2014}}</ref>
+ | year = 2014| doi = 10.37236/4069
+ | doi-access = free
+ }}</ref>
-Apart from finding the minimum number ''d'' of dimensions such that the number of pieces <math>\alpha(d) > d+1</math> mathematicians are interested in finding the general behavior of the function <math>\alpha(d)</math>. Kahn and Kalai show that in general (that is for ''d'' big enough), one needs <math>\alpha(d) \ge (1.2)^\sqrt{d}</math> number of pieces. They also quote the upper bound by [[Oded Schramm]], who showed that for every ''ε'', if ''d'' is sufficiently large, <math>\alpha(d) \le \left(\sqrt{3/2} + \varepsilon\right)^d</math>.<ref>{{citation
+Apart from finding the minimum number {{mvar|n}} of dimensions such that the number of pieces {{math|''α''(''n'') > ''n'' + 1}}, mathematicians are interested in finding the general behavior of the function {{math|''α''(''n'')}}. Kahn and Kalai show that in general (that is, for {{mvar|n}} sufficiently large), one needs <math display="inline">\alpha(n) \ge (1.2)^\sqrt{n}</math> many pieces. They also quote the upper bound by [[Oded Schramm]], who showed that for every {{mvar|ε}}, if {{mvar|n}} is sufficiently large, <math display="inline">\alpha(n) \le \left(\sqrt{3/2} + \varepsilon\right)^n</math>.<ref>{{citation
  | last = Schramm | first = Oded | authorlink1 = Oded Schramm
  | mr = 0986627
- | journal = Mathematika
+ | journal = [[Mathematika]]
  | pages = 180–189
  | title = Illuminating sets of constant width
@@ Line 128: / Line 155: @@
  | issue = 2
  | year = 1988
- | doi = 10.1112/S0025579300015175}}</ref> The correct order of magnitude of ''α''(''d'') is still unknown.<ref>{{citation
+ | doi = 10.1112/S0025579300015175}}</ref> The correct order of magnitude of {{math|''α''(''n'')}} is still unknown.<ref>{{citation
  | last = Alon | first = Noga | authorlink1 = Noga Alon
- | journal = Proceedings of the [[International Congress of Mathematicians]], [[Beijing]]
+ | journal = Proceedings of the International Congress of Mathematicians, Beijing
  | pages = 119–135
  | title = Discrete mathematics: methods and challenges
  | volume = 1
  | year = 2002
- | arxiv = math/0212390}}</ref> However, it is conjectured that there is a constant {{nobr|''c'' > 1}} such that <math>\alpha(d) > c^d</math> for all {{nobr|''d'' ≥ 1}}.
+ | arxiv = math/0212390| bibcode = 2002math.....12390A}}</ref> However, it is conjectured that there is a constant {{math|''c'' > 1}} such that {{math|''α''(''n'') > ''c<sup>n</sup>''}} for all {{math|''n'' ≥ 1}}.
 ==See also==
-*[[Hadwiger conjecture (combinatorial geometry)|Hadwiger's conjecture]] on covering convex bodies with smaller copies of themselves
+*[[Hadwiger conjecture (combinatorial geometry)|Hadwiger's conjecture]] on covering convex fields with smaller copies of themselves
+* [[Kahn–Kalai conjecture]]
 ==Note==
@@ Line 145: / Line 173: @@
 ==References==
 {{reflist|refs=
-<ref name=HinrRicht>{{cite journal| first1=Aicke| last1=Hinrichs| first2=Christian| last2=Richter| title=New sets with large Borsuk numbers| journal=[[Discrete Mathematics (journal)|Discrete Mathematics]]| volume=270| issue=1–3| date=28 August 2003| p=137&ndash;147| publisher=[[Elsevier]]| doi=10.1016/S0012-365X(02)00833-6}}
+<ref name=HinrRicht>{{cite journal| first1=Aicke| last1=Hinrichs| first2=Christian| last2=Richter| title=New sets with large Borsuk numbers| journal=[[Discrete Mathematics (journal)|Discrete Mathematics]]| volume=270| issue=1–3| pages=137–147| date=28 August 2003| publisher=[[Elsevier]]| doi=10.1016/S0012-365X(02)00833-6| doi-access=free}}
 </ref>
 }}
 ==Further reading==
-* Oleg Pikhurko, ''[http://www.math.cmu.edu/~pikhurko/AlgMet.ps Algebraic Methods in Combinatorics]'', course notes.
+* Oleg Pikhurko, ''[https://web.archive.org/web/20160304000819/http://www.math.cmu.edu/~pikhurko/AlgMet.ps Algebraic Methods in Combinatorics]'', course notes.
 * Andrei M. Raigorodskii, The Borsuk partition problem: the seventieth anniversary, ''[[Mathematical Intelligencer]]'' '''26''' (2004), no. 3, 4&ndash;12.
 * {{cite book | last=Raigorodskii | first=Andreii M. | chapter=Three lectures on the Borsuk partition problem | zbl=1144.52005 | editor1-last=Young | editor1-first=Nicholas | editor2-last=Choi | editor2-first=Yemon | title=Surveys in contemporary mathematics | publisher=[[Cambridge University Press]] | isbn=978-0-521-70564-6 | series=London Mathematical Society Lecture Note Series | volume=347 | pages=202–247 | year=2008 }}