Editing Ramsey's theorem (section)

== Ramsey numbers ==
<!-- This section is linked from [[Unsolved problems in mathematics]] -->
The numbers {{math|''R''(''r'', ''s'')}} in Ramsey's theorem (and their extensions to more than two colours) are known as Ramsey numbers. The Ramsey number {{math|''R''(''m'', ''n'')}} gives the solution to the party problem, which asks the minimum number of guests, {{math|''R''(''m'', ''n'')}}, that must be invited so that at least {{mvar|m}} will know each other or at least {{mvar|n}} will not know each other. In the language of graph theory, the Ramsey number is the minimum number of vertices, {{math|1=''v'' = ''R''(''m'', ''n'')}}, such that all undirected simple graphs of order {{mvar|v}}, contain a clique of order {{mvar|m}}, or an independent set of order {{mvar|n}}. Ramsey's theorem states that such a number exists for all {{mvar|m}} and {{mvar|n}}.

By symmetry, it is true that {{math|1=''R''(''m'', ''n'') = ''R''(''n'', ''m'')}}. An upper bound for {{math|''R''(''r'', ''s'')}} can be extracted from the proof of the theorem, and other arguments give lower bounds. (The first exponential lower bound was obtained by [[Paul Erdős]] using the [[probabilistic method]].) However, there is a vast gap between the tightest lower bounds and the tightest upper bounds. There are also very few numbers {{mvar|r}} and {{mvar|s}} for which we know the exact value of {{math|''R''(''r'', ''s'')}}.

Computing a lower bound {{mvar|L}} for {{math|''R''(''r'', ''s'')}} usually requires exhibiting a blue/red colouring of the graph {{math|''K''{{sub|''L''−1}}}} with no blue {{mvar|K{{sub|r}}}} subgraph and no red {{mvar|K{{sub|s}}}} subgraph. Such a counterexample is called a ''Ramsey graph''. [[Brendan McKay (mathematician)|Brendan McKay]] maintains a list of known Ramsey graphs.<ref name=McKay>{{cite web|url=http://cs.anu.edu.au/~bdm/data/ramsey.html|title=Ramsey Graphs}}</ref> Upper bounds are often considerably more difficult to establish: one either has to check all possible colourings to confirm the absence of a counterexample, or to present a mathematical argument for its absence.

=== Computational complexity ===
{{blockquote|text=[[Paul Erdős|Erdős]] asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of {{math|''R''(5, 5)}} or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for {{math|''R''(6, 6)}}. In that case, he believes, we should attempt to destroy the aliens.<ref>{{citation|title=Ten Lectures on the Probabilistic Method|page=[https://archive.org/details/tenlecturesonpro0000spen/page/4 4]|author=Joel H. Spencer|author-link=Joel H. Spencer|year=1994|publisher=[[Society for Industrial and Applied Mathematics|SIAM]]|isbn=978-0-89871-325-1|url=https://archive.org/details/tenlecturesonpro0000spen/page/4}}</ref>|author=[[Joel Spencer]]}}

A sophisticated computer program does not need to look at all colourings individually in order to eliminate all of them; nevertheless it is a very difficult computational task that existing software can only manage on small sizes. Each complete graph {{mvar|K{{sub|n}}}} has {{math|{{sfrac|1|2}}''n''(''n'' − 1)}} edges, so there would be a total of {{math|''c''{{sup|''n''(''n'' − 1)/2}}}} graphs to search through (for {{math|''c''}} colours) if brute force is used.<ref>[http://www.learner.org/channel/courses/mathilluminated/units/2/textbook/06.php 2.6 Ramsey Theory from Mathematics Illuminated]</ref> Therefore, the complexity for searching all possible graphs (via [[brute-force search|brute force]]) is {{math|''O''(''c''{{sup|''n''{{sup|2}}}})}} for {{mvar|c}} colourings and at most {{mvar|n}} nodes.

The situation is unlikely to improve with the advent of [[quantum computer]]s.  One of the best-known searching algorithms for unstructured datasets exhibits only a quadratic speedup (cf. [[Grover's algorithm]]) relative to classical computers, so that the [[Time complexity|computation time]] is still [[Exponential growth|exponential]] in the number of nodes.<ref>{{Cite journal |last=Montanaro |first=Ashley |date=2016 |title=Quantum algorithms: an overview |url=https://www.nature.com/articles/npjqi201523 |journal=[[npj Quantum Information]] |volume=2 |issue=1 |page=15023 |doi=10.1038/npjqi.2015.23 |arxiv=1511.04206 |bibcode=2016npjQI...215023M |s2cid=2992738 |via=Nature}}</ref><ref>{{Cite journal|last1=Wang|first1=Hefeng|year=2016|title=Determining Ramsey numbers on a quantum computer|journal=Physical Review A|volume=93|issue=3|pages=032301|arxiv=1510.01884|bibcode=2016PhRvA..93c2301W|doi=10.1103/PhysRevA.93.032301|s2cid=118724989}}</ref>

=== Known values ===
{{unsolved|mathematics|What is the value of {{math|''R''(5, 5)}}?}}
As described above, {{math|1=''R''(3, 3) = 6}}. It is easy to prove that {{math|1=''R''(4, 2) = 4}}, and, more generally, that {{math|1=''R''(''s'', 2) = ''s''}} for all {{mvar|s}}: a graph on {{math|''s'' − 1}} nodes with all edges coloured red serves as a counterexample and proves that {{math|''R''(''s'', 2) ≥ ''s''}}; among colourings of a graph on {{mvar|s}} nodes, the colouring with all edges coloured red contains a {{mvar|s}}-node red subgraph, and all other colourings contain a 2-node blue subgraph (that is, a pair of nodes connected with a blue edge.)

Using induction inequalities and the [[handshaking lemma]], it can be concluded that {{math|1=''R''(4, 3) ≤ ''R''(4, 2) + ''R''(3, 3) − 1 = 9}}, and therefore {{math|''R''(4, 4) ≤ ''R''(4, 3) + ''R''(3, 4) ≤ 18}}. There are only two {{math|(4, 4, 16)}} graphs (that is, 2-colourings of a complete graph on 16 nodes without 4-node red or blue complete subgraphs) among {{nowrap|6.4 × 10{{sup|22}}}} different 2-colourings of 16-node graphs, and only one {{math|(4, 4, 17)}} graph (the [[Paley graph]] of order 17) among {{nowrap|2.46 × 10{{sup|26}}}} colourings.<ref name=McKay /> It follows that {{math|1=''R''(4, 4) = 18}}.

The fact that {{math|1=''R''(4, 5) = 25}} was first established by [[Brendan McKay (mathematician)|Brendan McKay]] and [[Stanisław Radziszowski]] in 1995.<ref name=McKay1995>{{cite journal|title=''R''(4,5) = 25|last1=McKay | first1=Brendan D. | last2=Radziszowski | first2=Stanislaw P.|journal=[[Journal of Graph Theory]]|date=May 1995|doi=10.1002/jgt.3190190304|volume=19|issue=3|pages=309–322|url = http://users.cecs.anu.edu.au/~bdm/papers/r45.pdf}}</ref>

The exact value of {{math|''R''(5, 5)}} is unknown, although it is known to lie between 43 (Geoffrey Exoo (1989)<ref>{{cite journal |last1=Exoo |first1=Geoffrey |title=A lower bound for {{math| ''R''(5, 5)}} |journal=[[Journal of Graph Theory]] |date=March 1989 |volume=13 |issue=1 |pages=97–98 |doi=10.1002/jgt.3190130113}}</ref>) and 46 (Angeltveit and McKay (2024)<ref name=AngeltveitandMcKay2024>{{cite arXiv |eprint=2409.15709 |class=math.CO |author1=Vigleik Angeltveit |author2=Brendan McKay|title=<math>R(5,5) \leq 46</math> |date=September 2024}}</ref>), inclusive.

In 1997, McKay, Radziszowski and Exoo employed computer-assisted graph generation methods to conjecture that {{math|1=''R''(5, 5) = 43}}. They were able to construct exactly 656 {{math|(5, 5, 42)}} graphs, arriving at the same set of graphs through different routes. None of the 656 graphs can be extended to a {{math|(5, 5, 43)}} graph.<ref>{{cite journal |url=http://cs.anu.edu.au/~bdm/papers/r55.pdf| title=Subgraph Counting Identities and Ramsey Numbers |author=Brendan D. McKay, Stanisław P. Radziszowski |journal=[[Journal of Combinatorial Theory]] |series=Series B |doi=10.1006/jctb.1996.1741 |volume=69| issue=2|pages=193–209| year=1997|doi-access=free}}</ref>

For {{math|''R''(''r'', ''s'')}} with {{math|''r'', ''s'' > 5}}, only weak bounds are available. Lower bounds for {{math|''R''(6, 6)}} and {{math|''R''(8, 8)}} have not been improved since 1965 and 1972, respectively.<ref name="bananas" />

{{math|''R''(''r'', ''s'')}} with {{math|''r'', ''s'' ≤ 10}} are shown in the table below. Where the exact value is unknown, the table lists the best known bounds. {{math|''R''(''r'', ''s'')}} with {{math|''r'' < 3}} are given by {{math|1=''R''(1, ''s'') = 1}} and {{math|1=''R''(2, ''s'') = ''s''}} for all values of {{mvar|s}}.

The standard survey on the development of Ramsey number research is the ''Dynamic Survey 1'' of the ''[[Electronic Journal of Combinatorics]]'', by [[Stanisław Radziszowski]], which is periodically updated.<ref name="bananas">{{cite journal | url = http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS1/pdf | title = Small Ramsey Numbers|doi=10.37236/21|doi-access=free|journal=Electronic Journal of Combinatorics|department=Dynamic Surveys|date=2011|last1=Radziszowski|first1=Stanisław| volume = 1000|author1-link=Stanisław Radziszowski}}</ref><ref name="sprSRS">{{cite web|url=https://www.cs.rit.edu/~spr/ElJC/eline.html|title=DS1|author=Stanisław Radziszowski|access-date=17 August 2023}}</ref>   Where not cited otherwise, entries in the table below are taken from the June 2024 edition.  (Note there is a trivial symmetry across the diagonal since {{math|1=''R''(''r'', ''s'') = ''R''(''s'', ''r'')}}.)

{| class="wikitable"
|+ Values / known bounding ranges for Ramsey numbers {{math|''R''(''r'', ''s'')}} {{OEIS|id=A212954}}
! {{diagonal split header|''r''|''s''}}
! width="27" | 1
! width="33" | 2
! width="39" | 3
! width="45" | 4
! 5
! 6
! 7
! 8
! 9
! 10
|-
! 1
| {{yes|'''1'''}}
| {{yes|1}}
| {{yes|1}}
| {{yes|1}}
| {{yes|1}}
| {{yes|1}}
| {{yes|1}}
| {{yes|1}}
| {{yes|1}}
| {{yes|1}}
|-
! 2
|
| {{yes|'''2'''}}
| {{yes|3}}
| {{yes|4}}
| {{yes|5}}
| {{yes|6}}
| {{yes|7}}
| {{yes|8}}
| {{yes|9}}
| {{yes|10}}
|-
! 3
|
|
| {{yes|'''6'''}}
| {{yes|9}}
| {{yes|14}}
| {{yes|18}}
| {{yes|23}}
| {{yes|28}}
| {{yes|36}}
| {{partial|40–41<ref name=Angeltveit2023>{{cite arXiv |eprint=2401.00392 |class=math.CO |first1=Vigleik |last1=Angeltveit|title=<math>R(3,10) \leq 41</math> |date=31 Dec 2023}}</ref>}}
|-
! 4
|
|
|
| {{yes|'''18'''}}
| {{yes|25<ref name=McKay1995 />}}
| {{partial|36–40}}
| {{partial|49–58}}
| {{partial|59<ref name="Exoo2015">{{cite journal | journal = [[Electronic Journal of Combinatorics]] | volume = 22 | issue = 3 | year = 2015 | title = New Lower Bounds for 28 Classical Ramsey Numbers | last1 = Exoo | first1 = Geoffrey | last2 = Tatarevic | first2 = Milos | page = 3 | url = http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p11| arxiv = 1504.02403 |doi=10.37236/5254 |doi-access=free}}</ref>–79}}
| {{partial|73–105}}
| {{partial|92–135}}
|-
! 5
|
|
|
|
| {{partial|'''43–46'''<ref name=AngeltveitandMcKay2024>{{cite arXiv |eprint=2409.15709 |class=math.CO |author1=Vigleik Angeltveit |author2=Brendan McKay|title=<math>R(5,5) \leq 46</math> |date=September 2024}}</ref>}}
| {{partial|59<ref name="Exoo2023">{{cite arXiv |eprint=2310.17099 |class=math.CO |first1=Geoffrey |last1=Exoo |title=A Lower Bound for R(5,6) |date=26 Oct 2023}}</ref>–85}}
| {{partial|80–133}}
| {{partial|101–193}}
| {{partial|133–282}}
| {{partial|149<ref name="Exoo2015" />–381}}
|-
! 6
|
|
|
|
|
| {{partial|'''102–160'''}}
| {{partial|115<ref name="Exoo2015" />–270}}
| {{partial|134<ref name="Exoo2015" />–423}}
| {{partial|183–651}}
| {{partial|204–944}}
|-
! 7
|
|
|
|
|
|
| {{partial|'''205–492'''}}
| {{partial|219–832}}
| {{partial|252–1368}}
| {{partial|292–2119}}
|-
! 8
|
|
|
|
|
|
|
| {{partial|'''282–1518'''}}
| {{partial|329–2662}}
| {{partial|343–4402}}
|-
! 9
|
|
|
|
|
|
|
|
| {{partial|'''565–4956'''}}
| {{partial|581–8675}}
|-
! 10
|
|
|
|
|
|
|
|
|
| {{partial|'''798–16064'''}}
|}

It is also interesting that Erdos showed  
R(''P''{{sub|n}}, ''K''{{sub|m}})  = (n − 1).(m − 1) + 1,
for a path and a complete graph with n and m vertices respectively. Also Chvatal showed 
R(''T''{{sub|n}}, ''K''{{sub|m}})  = (n − 1).(m − 1) + 1,
for a tree and a complete graph with n and m vertices respectively. These two theorems are the best examples of formulating Ramsey numbers for some special graphs.

=== Asymptotics ===
The inequality {{math|''R''(''r'', ''s'') ≤ ''R''(''r'' − 1, ''s'') + ''R''(''r'', ''s'' − 1)}} may be applied inductively to prove that
:<math>R(r, s) \leq \binom{r + s - 2}{r - 1}. </math>

In particular, this result, due to [[Paul Erdős|Erdős]] and [[George Szekeres|Szekeres]], implies that when {{math|1=''r'' = ''s''}},
:<math>R(s, s) \leq (1 + o(1))\frac{4^{s - 1}}{\sqrt{\pi s}}.</math>

An exponential lower bound,
:<math>R(s, s) \geq (1 + o(1)) \frac{s}{\sqrt{2} e} 2^{s/2},</math>

was given by Erdős in 1947 and was instrumental in his introduction of the probabilistic method. There is a huge gap between these two bounds: for example, for {{math|1=''s'' = 10}}, this gives {{math|101 ≤ ''R''(10, 10) ≤ 48,620}}. Nevertheless, the exponential growth factors of either bound were not improved for a long time, and for the lower bound it still stands at {{math|{{sqrt|2}}}}. There is no known explicit construction producing an exponential lower bound. The best known lower and upper bounds for diagonal Ramsey numbers are
:<math>[1 + o(1)] \frac{\sqrt{2} s}{e} 2^{\frac{s}{2}} \leq R(s, s) \leq s^{-(c \log s)/(\log \log s)} 4^s,</math>

due to [[Joel Spencer|Spencer]] and [[David Conlon|Conlon]], respectively; a 2023 preprint by Campos, Griffiths, [[Robert Morris (mathematician)|Morris]] and [[Julian Sahasrabudhe|Sahasrabudhe]] claims to have made exponential progress using an algorithmic construction relying on a graph structure called a "[[Book (graph theory)|book]]",<ref name="arxiv23">{{cite arXiv |eprint=2303.09521 |class=math.CO |first1=Marcelo |last1=Campos |first2=Simon |last2=Griffiths |title=An exponential improvement for diagonal Ramsey |last3=Morris |first3=Robert |last4=Sahasrabudhe |first4=Julian |year=2023}}</ref><ref>{{Cite web |last=Sloman |first=Leila |date=2 May 2023 |title=A Very Big Small Leap Forward in Graph Theory |url=https://www.quantamagazine.org/after-nearly-a-century-a-new-limit-for-patterns-in-graphs-20230502/ |website=[[Quanta Magazine]]}}</ref> improving the upper bound to
:<math>R(s,s) \leq (4-\varepsilon)^{s}\text{ and } R(s,t) \leq e^{-\delta t+o(s)}\binom{s+t}{t}.</math>
with <math>\varepsilon=2^{-7}</math> and <math>\delta=50^{-1}</math>.

A 2024 preprint<ref>{{cite arXiv |last1=Gupta |first1=Parth |title=Optimizing the CGMS upper bound on Ramsey numbers |date=2024-07-26 |eprint=2407.19026 |last2=Ndiaye |first2=Ndiame |last3=Norin |first3=Sergey |last4=Wei |first4=Louis|class=math.CO }}</ref> by Gupta, Ndiaye, Norin, and Wei claims an improvement of <math>\delta </math> to <math>-0.14e^{-1} \leq 20^{-1}</math>, and the diagonal Ramsey upper bound to

<math>R(s,s) \leq \left(4e^{-0.14e^{-1}}\right)^{s + o(s)} = 3.7792...^{s+o(s)}</math>

For the off-diagonal Ramsey numbers {{math|''R''(3, ''t'')}}, it is known that they are of order {{math|{{sfrac|''t''{{sup|2}}|log ''t''}}}}; this may be stated equivalently as saying that the smallest possible [[Independent set (graph theory)|independence number]] in an {{mvar|n}}-vertex [[triangle-free graph]] is
:<math>\Theta \left (\sqrt{n\log n} \right ).</math>

The upper bound for {{math|''R''(3, ''t'')}} is given by [[Miklós Ajtai|Ajtai]], [[János Komlós (mathematician)|Komlós]], and [[Endre Szemerédi|Szemerédi]],<ref>{{Cite journal |last1=Ajtai |first1=Miklós |last2=Komlós |first2=János |last3=Szemerédi |first3=Endre |date=1980-11-01 |title=A note on Ramsey numbers |journal=Journal of Combinatorial Theory, Series A |language=en |volume=29 |issue=3 |pages=354–360 |doi=10.1016/0097-3165(80)90030-8 |issn=0097-3165|doi-access=free }}</ref> the lower bound was obtained originally by [[Jeong Han Kim|Kim]],<ref>{{Citation |last1=Kim |first1=Jeong Han |title=The Ramsey Number ''R''(3,''t'') has order of magnitude ''t''<sup>2</sup>/log ''t'' |journal=Random Structures and Algorithms |volume=7 |issue=3 |pages=173–207 |year=1995 |citeseerx=10.1.1.46.5058 |doi=10.1002/rsa.3240070302 |author-link1=<!--Jeong Han Kim-->}}</ref> and the implicit constant was improved independently by Fiz Pontiveros, Griffiths and [[Robert Morris (mathematician)|Morris]],<ref>{{Cite web |title=The Triangle-Free Process and the Ramsey Number {{math|''R''(3,''k'')}} |url=https://bookstore.ams.org/memo-263-1274 |access-date=2023-06-27 |website=bookstore.ams.org}}</ref> and [[Tom Bohman|Bohman]] and [[Peter Keevash|Keevash]],<ref>{{Cite journal |last1=Bohman |first1=Tom |last2=Keevash |first2=Peter |date=2020-11-17 |title=Dynamic concentration of the triangle-free process |url=https://doi.org/10.1002/rsa.20973 |journal=Random Structures and Algorithms |language=en |volume=58 |issue=2 |pages=221–293 |doi=10.1002/rsa.20973 | arxiv=1302.5963 }}</ref> by analysing the triangle-free process. 

In general, studying the more general "{{math|''H''}}-free process" has set the best known asymptotic lower bounds for general off-diagonal Ramsey numbers,<ref>{{Cite journal |last1=Bohman |first1=Tom |last2=Keevash |first2=Peter |date=2010-08-01 |title=The early evolution of the H-free process |url=https://doi.org/10.1007/s00222-010-0247-x |journal=Inventiones Mathematicae |language=en |volume=181 |issue=2 |pages=291–336 |doi=10.1007/s00222-010-0247-x |issn=1432-1297| arxiv=0908.0429|bibcode=2010InMat.181..291B }}</ref> {{math|''R''(''s'', ''t'')}}
:<math>c'_s \frac{t^{\frac{s + 1}{2}}}{(\log t)^{\frac{s + 1}{2} - \frac{1}{s - 2}}} \leq R(s, t) \leq c_s \frac{t^{s - 1}}{(\log t)^{s - 2}}.</math>

In particular this gives an upper bound of <math>R(4, t) \leq c_s t^3(\log t)^{-2}</math>. Mattheus and Verstraete (2024)<ref>{{Cite journal | journal=Annals of Mathematics |last1=Mattheus |first1=Sam |last2=Verstraete |first2=Jacques |date=5 Mar 2024 |title=The asymptotics of r(4,t) |volume=199 |issue=2 |arxiv=2306.04007 |doi=10.4007/annals.2024.199.2.8}}</ref><ref>{{Cite web |last=Cepelewicz |first=Jordana |date=22 June 2023 |title=Mathematicians Discover Novel Way to Predict Structure in Graphs |url=https://www.quantamagazine.org/mathematicians-discover-new-way-to-predict-structure-in-graphs-20230622/ |website=[[Quanta Magazine]]}}</ref> gave a lower bound of  <math>R(4, t) \geq c'_s t^3(\log t)^{-4}</math>, determining the asymptotics of <math>R(4, t)</math> up to logarithmic factors, and settling a question of Erdős, who offered 250 dollars for a proof that the lower limit has form <math>c'_s t^{3}(\log t)^{-d}</math>.<ref>{{Citation |last=Erdös |first=Paul |title=Problems and Results on Graphs and Hypergraphs: Similarities and Differences |date=1990 |url=https://doi.org/10.1007/978-3-642-72905-8_2 |work=Mathematics of Ramsey Theory |pages=12–28 |editor-last=Nešetřil |editor-first=Jaroslav |access-date=2023-06-27 |series=Algorithms and Combinatorics |volume=5 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-642-72905-8_2 |isbn=978-3-642-72905-8 |editor2-last=Rödl |editor2-first=Vojtěch}}</ref><ref>{{Cite web |title=Erdős Problems |url=https://www.erdosproblems.com/166 |access-date=2023-07-12 |website=www.erdosproblems.com}}</ref>

=== Formal verification of Ramsey numbers ===

The Ramsey number <math>R(3,8)</math> and <math>R(3,9)</math> have been formally verified to be 28 and 36.<ref>{{cite arXiv |last1=Li |first1=Zhengyu |last2=Duggan |first2=Conor |last3=Bright |first3=Curtis |last4=Ganesh |first4=Vijay |title=Verified Certificates via SAT and Computer Algebra Systems for the Ramsey R(3, 8) and R(3, 9) Problems |date=2025 |class=cs.LO |eprint=2502.06055}}</ref> This verification was achieved using a combination of Boolean satisfiability (SAT) solving and computer algebra systems (CAS). The proof was generated automatically using the [https://uwaterloo.ca/mathcheck/ SAT+CAS] approach, marking the first certifiable proof of <math>R(3,8) = 28</math> and <math>R(3,9)=36</math>. The verification process for <math>R(3,8)</math> and <math>R(3,9)</math> was conducted using the SAT+CAS framework MathCheck, which integrates a SAT solver with a computer algebra system. The verification for <math>R(3,8) = 28</math> was completed in approximately 8 hours of wall clock time, producing a total proof size of 5.8 GiB. The verification for <math>R(3,9) = 36</math> was significantly more computationally intensive, requiring 26 hours of wall clock time and generating 289 GiB of proof data. The correctness of these results was independently verified using a modified version of the [https://www.cs.utexas.edu/~marijn/drat-trim/ DRAT-trim] proof checker.<ref>{{cite arXiv |last1=Li |first1=Zhengyu |last2=Duggan |first2=Conor |last3=Bright |first3=Curtis |last4=Ganesh |first4=Vijay |title=Verified Certificates via SAT and Computer Algebra Systems for the Ramsey R(3, 8) and R(3, 9) Problems |date=2025 |class=cs.LO |eprint=2502.06055}}</ref>

The Ramsey number <math>R(4,5)</math> has been formally verified to be 25.<ref>{{cite arXiv |last1=Gauthier |first1=Thibault |last2=Brown |first2=Chad E |title=A formal proof of R(4,5)=25 |date=2024 |class=cs.LO |eprint=2404.01761}}</ref> The original proof, developed by McKay and Radziszowski in 1995, combined high-level mathematical arguments with computational steps and used multiple independent implementations to reduce the possibility of programming errors. The formal proof was carried out using the [[HOL (proof assistant)|HOL4]] interactive theorem prover, limiting the potential for errors to the HOL4 kernel. Rather than directly verifying the original algorithms, the authors utilized HOL4's interface to the MiniSat SAT solver to formally prove key gluing lemmas.