Pith. sign in

REVIEW 1 major objections 5 minor 1 cited by

A Survey on Ordered Ramsey Numbers

T0 review · 1 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper surveys ordered Ramsey numbers and reports that imposing a linear order on vertices can inflate Ramsey numbers far beyond the unordered case, while identifying structural parameters that restore polynomial bounds.

desk verdict A readable survey with real organizing value, but one load-bearing typo in Theorem 14 and two unpublished references keep it from being a reliable reference as-is. read the letter →

arxiv 2502.02155 v1 pith:MEIZAXN5 submitted 2025-02-04 math.CO cs.DM

Martin Balko This is my paper · ORCID
classification math.COcs.DM MSC 05D1005C5505C65
keywords orderedRamseynumberstheorygraphsintervalchromaticnumberbandwidthmonotonepathsedge-orderedErdős–Szekerestheorem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Ordered Ramsey numbers ask how large a complete graph with linearly ordered vertices must be before every red-blue edge coloring contains a monochromatic copy of a given ordered graph in the prescribed order. The survey's thesis is that imposing this order changes the quantitative story: sparse ordered graphs such as matchings can have superpolynomial ordered Ramsey numbers, whereas unordered bounded-degree graphs have linear Ramsey numbers. It gathers the main upper and lower bounds, highlights parameters such as interval chromatic number and bandwidth that restore polynomial growth, records exact formulas tied to the Erdős–Szekeres theorem, and lists open problems spanning multicolor, hypergraph, and edge-ordered variants.

What carries the argument

The load-bearing objects are the ordered Ramsey number R_<(G_<,H_<), defined as the smallest N such that every two-coloring of the ordered complete graph K_N^< yields a red copy of G_< or a blue copy of H_<, and the structural parameters used to bound it: interval chromatic number (the fewest intervals into which the vertex set can be partitioned so no edge lies inside an interval), degeneracy, and bandwidth (longest edge length). The named example carrying the connection to classical geometry is the monotone path MP_n^<, whose exact ordered Ramsey number (n-1)^2+1 is equivalent to the Erdős–Szekeres lemma on monotone subsequences.

What would settle it

A concrete check: once the in-preparation and submitted references appear, verify whether their stated bounds still hold; if either fails, the corresponding survey sections on multicolor ordered matchings and the Erdős–Szekeres reformulation would be wrong. Alternatively, a single ordered graph of bandwidth 2 with ordered Ramsey number exceeding C $n^{{4+ε}}$ would disprove the strongest theorem the survey reports.

Watch

Extended reading notes

Core claim

The author's aim is to establish that ordered Ramsey numbers form a subject with its own quantitative laws, distinct from classical Ramsey numbers. The survey reports that sparse ordered graphs can be extremely Ramsey-hard—there exist ordered matchings on n vertices with R_<(M_<) ≥ $n^{{C log n / log log n}}$—while bounded interval chromatic number and bounded degeneracy restore polynomial growth, R_<(G_<) ≤ $n^{{32 d log χ}}$. For monotone paths and cycles the survey records essentially sharp or exact results, including R_<(MC_r^<, MC_s^<) = 2rs - 3r - 3s + 6 and the monotone path formula that underlies the Erdős–Szekeres lemma. It presents the current best bound R_<(P_{k,n}^<) ≤ C $n^{{4+ε}}$ for bounded-bandwidth graphs and a list of open problems that mark where the subject's limits sit.

Load-bearing premise

The survey's picture is only as reliable as the cited results, and several of them have not yet been published, so a reader cannot yet independently verify those pieces.

Editorial extensions

If this is right

  • If the reported bounds are correct, every ordered graph with fixed interval chromatic number and fixed degeneracy has a polynomial ordered Ramsey number, so Ramsey-type arguments for such ordered graphs do not need to pass through exponential bounds.
  • The exact monotone-cycle formula transfers directly to convex and geometric Ramsey numbers, giving R_c(C_n) = R_g(C_n) = 2n^2 - 6n + 6 for cycles.
  • The superpolynomial lower bound for ordered matchings means that ordered Ramsey theory is genuinely different from unordered Ramsey theory even for sparse graphs.
  • A positive answer to the survey's open problem about powers of monotone paths would replace the n^{4+o(1)} bound by a near-quadratic one, sharpening the whole bounded-bandwidth picture.
  • The survey's open problems, especially the off-diagonal matching-versus-triangle problem, identify where known techniques stop and mark concrete targets for new constructions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One step beyond the paper: interval chromatic number may be the operative parameter for algorithmic ordered Ramsey computations, since degree alone is insufficient; a testable extension would be to compute ordered Ramsey numbers of random ordered graphs with interval chromatic number 2 and growing degree.
  • The near-quartic upper bound versus quadratic lower bound for powers of monotone paths suggests the true growth is closer to quadratic, as the cited authors conjecture; if so, all bounded-bandwidth ordered Ramsey numbers would be near-quadratic in n.
  • The connection between monotone hyperpaths and antichains suggests a new route to the Erdős–Szekeres conjecture: improving upper bounds on R_<(MP_n^{(3)}) is equivalent to an enumerative problem on antichains, which could be attacked computationally.
  • The edge-ordered section points toward a wider principle: when the order is on edges rather than vertices, finiteness is non-trivial and requires completely different arguments; extending the exponential-type bounds to edge-ordered hypergraphs would be a natural next test.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. This survey synthesizes the recent literature on ordered Ramsey numbers. Section 2 covers ordered graphs: general bounds via degeneracy and interval chromatic number (Theorems 1-3), bounded-bandwidth paths (Theorem 5), off-diagonal matchings versus triangles, minimum ordered Ramsey numbers (Theorem 6), exact formulas for alternating paths and monotone cycles (Proposition 8, Theorem 9), and multicolor extensions. Section 3 covers ordered hypergraphs: monotone paths and the Erdos-Szekeres connection (Theorems 11-13), a relation between ordered Ramsey numbers and classical hypergraph Ramsey numbers (Theorem 14), and bounds for sparse 3-uniform hypergraphs (Theorems 16-17). Section 4 covers edge-ordered Ramsey numbers (Theorems 19-21). The paper compiles 16 open problems and 5 conjectures with attributions, and it explicitly disclaims exhaustiveness. No proofs are given; all statements are attributed to the literature.

Significance. If the reported results are accurate, this is a useful and timely reference: it appears to be the only recent survey devoted specifically to ordered Ramsey numbers, it covers the graph, hypergraph, and edge-ordered settings in one place, and it gives an attributed catalogue of open problems (Problems 1-16, Conjectures 1-5) that researchers can use directly. The author is a principal contributor to the area, and the bibliography is current through 2024. The connections drawn (Erdos-Szekeres theorem, geometric Ramsey numbers, k-queue graphs, Dedekind numbers) are presented clearly. Because the survey contains no derivations, its value rests entirely on the reliability of its attributions and displayed formulas; at present that reliability is compromised by at least one load-bearing misstatement (Theorem 14) and by the use of two non-public references ([3], [6]), so the manuscript needs revision before it can serve as the citation of record.

major comments (1)
  1. [3.2, Theorem 14] Theorem 14 is internally inconsistent and, as printed, false. The displayed statement is R(K^{(k-1)}_{floor(n/q)}; q) <= R_<(K^{(k)}_n, MP^{(k)}_{k-q+1}) <= R(K^{(k-1)}_n; q), while the immediately preceding paragraph says that the case q=2 concerns R_<(K^{(k)}_n, MP^{(k)}_{k+1}). For q=2 the printed index gives k-q+1 = k-1, so the middle term involves MP^{(k)}_{k-1}, a monotone path on k-1 vertices with no edges; the middle quantity is then trivially at most k-1 (for N = k-1 any coloring contains a blue copy), and the lower bound would assert R(K^{(k-1)}_{floor(n/2)}; 2) <= k-1, which is false (e.g., k=3 and n>=6 gives R(K^{(2)}_3; 2) = R(3,3) = 6 <= 2). The reading consistent with the surrounding text and with Conjecture 4 is MP^{(k)}_{k+q-1} (a monotone path with q edges), and the statement must quantify q >= 2 as well. Please correct the index (verifying the exact statement against [64]) and the quantifiers; as written, this advertised 'surprising relation' between ordered and classical hypergraph Ramsey numbers cannot be used by a reader.
minor comments (5)
  1. [2.1; 2.6; 3; refs] Typographical and name errors: 'Colon, Fox, Lee, and Sudakov' (Section 2.1, before Problem 3) should be 'Conlon, Fox, Lee, and Sudakov'; 'mathchings' (Section 2.6) should be 'matchings'; 'dentote' (Section 3, first paragraph) should be 'denote'; reference [3] gives 'revisitied' for 'revisited'; reference [46] lists 'Neeidinger' for 'Neidinger'.
  2. [2.2] In the paragraph after Theorem 5, the constant 'c = c(t)' should be 'c = c(k)', and the sentence describing the bound as valid 'for all s, t and n' uses an undefined parameter t (presumably k). The notation K_<^n in R_<(P_{k,n}, K_<^n) should be rendered as K^<_n, the ordered complete graph on n vertices.
  3. [2.3] The lower bound 'R_<(M_<, K_3) >= O((n/log n)^{4/3})' for ordered matchings should use Omega, not O, as the notation '>= O(...)' is meaningless as printed.
  4. [3.2] In Conjecture 4, R_<(K^{(k)}, MP^{(k)}_{k+1}) is missing the subscript n on the clique and should read R_<(K^{(k)}_n, MP^{(k)}_{k+1}). Also, the opening sentence of Section 3.2 refers to 'the classical Ramsey numbers R(K^{(k-1)}_n)', whereas Theorem 14 involves q-color Ramsey numbers; the wording should be aligned.
  5. [2.6 and 3.1] Several stated results and open problems rest on non-public references: [6] ('In preparation, 2024') supports the theorem R_<(M_<; q) = n^{Theta(q)} for matchings with interval chromatic number 2 and Problems 10-13, while [3] ('Submitted, 2024') is cited in Section 3.1. Since a survey is meant to be a stable reference, please replace these with publicly available preprints or published versions, or explicitly label the statements as conditional on forthcoming work.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the survey reports attributed external results and open problems; no derivation or prediction reduces to its own inputs.

full rationale

This is a survey paper, not a derivation. Its content consists of statements attributed to specific external papers (e.g., Theorems 1–21 are each credited to [27], [13], [48], [62], [64], [40], [44], etc.), and its open problems are posed as questions about the cited bounds rather than as outputs of a chain of reasoning contained in the paper. There are no fitted parameters, no quantities predicted from data within the paper, and no ansatz whose justification is imported from the authors' own prior work in a way that would force the conclusions. The closest structural relationship is the use of the exact formula R<(MP<^{(3)}_n) = binom(2n-4, n-2)+1 to derive the Erdős–Szekeres bound (7); this is presented as a consequence of the separately cited theorem of Moshkovitz and Shapira [62], which is external, parameter-free evidence and not equivalent to the Erdős–Szekeres theorem itself. Self-citations appear (e.g., [3], [6], [10], [11], [12], [13]), including two unpublished items [3] and [6], but they are used to attribute specific results or to point to problems; they do not carry a load-bearing argument that reduces to an unverified self-citation. The skeptical remark about Theorem 14 identifies a possible index error in a quoted result from [64]; even if correct, that is a correctness or typographical issue, not a circularity, since the survey adds no argument that defines its claimed connection in terms of itself. No exhibited reduction of the required kind (e.g., Eq. X equal to Eq. Y by construction, or a fitted parameter renamed as a prediction) exists in the paper, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

This is a survey paper with no new derivations, so it introduces no free parameters, axioms, or invented entities. The mathematical assumptions and theorems it relies on are all from the cited literature.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Survey on Ordered Ramsey Numbers." pith.science (2026). https://pith.science/paper/MEIZAXN5

@misc{pith2026250202155,
  author       = {Pith},
  title        = {Pith review of: A Survey on Ordered Ramsey Numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MEIZAXN5}},
  note         = {Machine review of arXiv:2502.02155}
}
abstract

The ordered Ramsey number of a graph $G^<$ with a linearly ordered vertex set is the smallest positive integer $N$ such that any two-coloring of the edges of the ordered complete graph on $N$ vertices contains a monochromatic copy of $G^<$ in the given ordering. The study of the quantitative behavior of ordered Ramsey numbers is a relatively new theme in Ramsey theory full of interesting and difficult problems. In this survey paper, we summarize recent developments in the theory of ordered Ramsey numbers. We point out connections to other areas of combinatorics and some well-known conjectures. We also list several new and challenging open problems and highlight the often strikingly different behavior from the unordered case.

Figures

Figures reproduced from arXiv: 2502.02155 by the authors.

Figure 1
Figure 1. Given a sequence S = (s1, . . . , sN ) of distinct real numbers, we construct an ordered graph K< n with vertex set S and the ordering of the vertices given by their positions in S. Then, we color an edge {si , sj} with i < j red if si < sj and blue otherwise. Afterward, red monotone paths on n vertices correspond to increasing subsequences of S of length n and blue monotone paths on n vertices to decreasing subsequ… view at source ↗
Figure 2
Figure 2. (a) The nested matching NMk for k = 3. (b) The alternating path AP < n for n = 7. (c) The monotone cycle MC< n for n = 7. Using a result from the extremal theory of {0, 1}-matrices, Balko, Cibulka, Kr´al, and Kynˇcl [13] proved the following linear bounds on R<(AP < n ). Proposition 8 ([13]). For every integer n > 2, we have 5⌊n/2⌋ − 4 ≤ R<(AP < n ) ≤ 2n − 3 + p 2n2 − 8n + 11. The proof of the lower bound was later … view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Computation of small reflective and dihedral Ramsey numbers

    math.CO 2026-07 accept novelty 5.5 of 10

    SAT-based computation yields exact small reflective and dihedral Ramsey numbers for several ordered graph families, plus closed formulas and conjectures linking them to ordered and cyclic variants.

Reference graph

Works this paper leans on

85 extracted references · 58 canonical work pages · cited by 1 Pith paper

  1. [6]

    Balko and K

    M. Balko and K. Grinerov´ a. Estimating multicolor ordered Ramsey numbers. In preparation, 2024

    2024

  2. [3]

    Baek and M

    J. Baek and M. Balko. The Erd˝ os–Szekeres conjecture revisitied. Submitted, 2024. 19

    2024

  3. [64]

    Mubayi and A

    D. Mubayi and A. Suk. Off-diagonal hypergraph Ramsey numbers. J. Combin. Theory Ser. B, 125:168–177, 2017. doi: 10.1016/j.jctb.2017.03.002

    work page doi:10.1016/j.jctb.2017.03.002 2017

  4. [1]

    Ajtai, J

    M. Ajtai, J. Koml´ os, and E. Szemer´ edi. A note on Ramsey numbers.J. Combin. Theory Ser. A, 29(3):354–360, 1980. doi: 10.1016/0097-3165(80)90030-8

    doi:10.1016/0097-3165(80)90030-8 1980

  5. [2]

    Alon and V

    N. Alon and V. R¨ odl. Sharp bounds for some multicolor Ramsey numbers.Combinatorica, 25(2):125–141, 2005. doi: 10.1007/s00493-005-0011-9

    work page doi:10.1007/s00493-005-0011-9 2005

  6. [4]

    Balister, B

    P. Balister, B. Bollob´ as, M. Campos, S. Griffiths, E. Hurley, R. Morris, J. Sahasrabudhe, and M. Tiba. Upper bounds for multicolour Ramsey numbers. http://arXiv.org/ abs/2410.17197, 2024

    arXiv 2024

  7. [5]

    M. Balko. Ramsey numbers and monotone colorings. J. Combin. Theory Ser. A , 163: 34–58, 2019. doi: 10.1016/j.jcta.2018.11.013

    work page doi:10.1016/j.jcta.2018.11.013 2019

  8. [7]

    Balko and M

    M. Balko and M. Poljak. On off-diagonal ordered Ramsey numbers of nested matchings. Discrete Math., 346(2):Paper No. 113223, 13, 2023. doi: 10.1016/j.disc.2022.113223

    work page arXiv 2023

Show all 85 references
  1. [8]

    Balko and M

    M. Balko and M. Poljak. On ordered Ramsey numbers of matchings versus triangles. Electron. J. Combin., 31(2):Paper No. 2.23, 16, 2024. doi: 10.37236/12107

    2024 doi

  2. [9]

    Balko and P

    M. Balko and P. Valtr. A SAT attack on the Erd˝ os–Szekeres conjecture.European J. Combin., 66:13–23, 2017. doi: 10.1016/j.ejc.2017.06.010

    2017 doi

  3. [10]

    Balko and M

    M. Balko and M. Vizer. Edge-ordered Ramsey numbers. European J. Combin. , 87: 103100, 11, 2020. doi: 10.1016/j.ejc.2020.103100

    2020

  4. [11]

    Balko and M

    M. Balko and M. Vizer. On ordered Ramsey numbers of tripartite 3-uniform hypergraphs. SIAM J. Discrete Math. , 36(1):214–228, 2022. doi: 10.1137/21M1404958

    2022 doi

  5. [12]

    Balko, V

    M. Balko, V. Jel´ ınek, and P. Valtr. On ordered Ramsey numbers of bounded-degree graphs. J. Combin. Theory Ser. B , 134:179–202, 2019. doi: 10.1016/j.jctb.2018.06.002

    2019 doi

  6. [13]

    Balko, J

    M. Balko, J. Cibulka, K. Kr´ al, and J. Kynˇ cl. Ramsey numbers of ordered graphs. Electron. J. Combin., 27:P1.16, 2020

    2020

  7. [14]

    Bar´ at, A

    J. Bar´ at, A. Gy´ arf´ as, and G. T´ oth. Monochromatic spanning trees and matchings in ordered complete graphs. J. Graph Theory, 105(4):523–541, 2024. doi: 10.1002/jgt.23058

    2024 doi

  8. [15]

    Campos, S

    M. Campos, S. Griffiths, R. Morris, and J. Sahasrabudhe. An exponential improvement for diagonal Ramsey. http://arXiv.org/abs/2303.09521, 2023

    2023 arXiv

  9. [16]

    S. A. Choudum and B. Ponnusamy. Ordered Ramsey numbers. Discrete Math., 247 (1-3):79–92, 2002. doi: 10.1016/S0012-365X(01)00161-3

    2002 doi

  10. [17]

    F. R. K. Chung and R. L. Graham. Forced convex n-gons in the plane. Discrete Comput. Geom., 19(3, Special Issue):367–371, 1998. doi: 10.1007/PL00009353. Dedicated to the memory of Paul Erd˝ os

    1998 doi

  11. [18]

    Chv´ atal, V

    V. Chv´ atal, V. R¨ odl, E. Szemer´ edi, and W. T. Trotter, Jr. The Ramsey number of a graph with bounded maximum degree. J. Combin. Theory Ser. B , 34(3):239–243, 1983. doi: 10.1016/0095-8956(83)90037-0

    1983 doi

  12. [19]

    Cibulka, P

    J. Cibulka, P. Gao, M. Krˇ c´ al, T. Valla, and P. Valtr. On the geometric Ramsey number of outerplanar graphs. Discrete Comput. Geom. , 53(1):64–79, 2015. doi: 10.1007/s00454-014-9646-x

    2015 doi

  13. [20]

    D. Conlon. A new upper bound for diagonal Ramsey numbers. Ann. of Math. (2) , 170 (2):941–960, 2009. doi: 10.4007/annals.2009.170.941. 20

    2009 doi

  14. [21]

    Conlon and A

    D. Conlon and A. Ferber. Lower bounds for multicolor Ramsey numbers. Adv. Math., 378:Paper No. 107528, 5, 2021. doi: 10.1016/j.aim.2020.107528

    2021

  15. [22]

    Conlon, J

    D. Conlon, J. Fox, and B. Sudakov. Ramsey numbers of sparse hypergraphs. Random Structures Algorithms, 35(1):1–14, 2009. doi: 10.1002/rsa.20260

    2009 doi

  16. [23]

    Conlon, J

    D. Conlon, J. Fox, and B. Sudakov. Hypergraph Ramsey numbers. J. Amer. Math. Soc., 23(1):247–266, 2010. doi: 10.1090/S0894-0347-09-00645-6

    2010 doi

  17. [24]

    Conlon, J

    D. Conlon, J. Fox, and B. Sudakov. Large almost monochromatic subsets in hypergraphs. Israel J. Math. , 181:423–432, 2011. doi: 10.1007/s11856-011-0016-6

    2011 doi

  18. [25]

    Conlon, J

    D. Conlon, J. Fox, and B. Sudakov. Erd˝ os–Hajnal-type theorems in hypergraphs. J. Combin. Theory Ser. B , 102(5):1142–1154, 2012. doi: 10.1016/j.jctb.2012.05.005

    2012 doi

  19. [26]

    Conlon, J

    D. Conlon, J. Fox, and B. Sudakov. Recent developments in graph Ramsey theory. In Surveys in combinatorics 2015 , volume 424 of London Math. Soc. Lecture Note Ser. , pages 49–118. Cambridge Univ. Press, Cambridge, 2015

    2015

  20. [27]

    Conlon, J

    D. Conlon, J. Fox, C. Lee, and B. Sudakov. Ordered Ramsey numbers. J. Combin. Theory Ser. B , 122:353–383, 2017. doi: 10.1016/j.jctb.2016.06.007

    2017 doi

  21. [28]

    Cooley, N

    O. Cooley, N. Fountoulakis, D. K¨ uhn, and Deryk Osthus. 3-uniform hypergraphs of bounded degree have linear Ramsey numbers. J. Combin. Theory Ser. B , 98(3):484–505,

  22. [29]

    Cooley, N

    O. Cooley, N. Fountoulakis, D. K¨ uhn, and D. Osthus. Embeddings and Ramsey numbers of sparse k-uniform hypergraphs. Combinatorica, 29(3):263–297, 2009. doi: 10.1007/s00493-009-2356-y

    2009 doi

  23. [30]

    Dujmovi´ c and D

    V. Dujmovi´ c and D. R. Wood. On linear layouts of graphs. Discrete Math. Theor. Comput. Sci., 6(2):339–357, 2004

    2004

  24. [31]

    Eli´ aˇ s and J

    M. Eli´ aˇ s and J. Matouˇ sek. Higher-order Erd˝ os–Szekeres theorems.Adv. Math., 244: 1–15, 2013. doi: 10.1016/j.aim.2013.04.020

    2013 doi

  25. [32]

    Erd˝ os and Vera T

    P. Erd˝ os and Vera T. S´ os. Problems and results on Ramsey-Tur´ an type theorems (preliminary report). In Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing (Humboldt State Univ., Arcata, Calif., 1979) , volume XXVI of Congress. Numer., pag...

    1979

  26. [33]

    P. Erd˝ os. Some remarks on the theory of graphs. Bull. Amer. Math. Soc. , 53:292–294,

  27. [34]

    Erd˝ os and A

    P. Erd˝ os and A. Hajnal. On Ramsey like theorems. Problems and results. In Combina- torics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972) , pages 123–140. Inst. Math. Appl., Southend-on-Sea, 1972

    1972

  28. [35]

    Erd˝ os and R

    P. Erd˝ os and R. Rado. Combinatorial theorems on classifications of subsets of a given set. Proc. London Math. Soc. (3) , 2:417–439, 1952. doi: 10.1112/plms/s3-2.1.417

    1952 doi

  29. [36]

    Erd˝ os and G

    P. Erd˝ os and G. Szekeres. A combinatorial problem in geometry.Compositio Math., 2: 463–470, 1935. 21

    1935

  30. [37]

    Erd˝ os and G

    P. Erd˝ os and G. Szekeres. On some extremum problems in elementary geometry. Ann. Univ. Sci. Budapest. E¨ otv¨ os Sect. Math., 3(4):53–62, 1960

    1960

  31. [38]

    Erd˝ os and A

    P. Erd˝ os and A. Szemer´ edi. On a Ramsey type theorem.Period. Math. Hungar. , 2: 295–299, 1972. doi: 10.1007/BF02018669

    1972 doi

  32. [39]

    Erd˝ os, A

    P. Erd˝ os, A. Hajnal, and R. Rado. Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar., 16:93–196, 1965. doi: 10.1007/BF01886396

    1965 doi

  33. [40]

    Falgas-Ravry, E

    V. Falgas-Ravry, E. R¨ aty, and I. Tomon. Dedekind’s problem in the hypergrid.http: //arXiv.org/abs/2310.12946, 2023

    2023 arXiv

  34. [41]

    Felsner and H

    S. Felsner and H. Weil. Sweeps, arrangements and signotopes. Discrete Appl. Math., 109(1-2):67–94, 2001. doi: 10.1016/S0166-218X(00)00232-8. 14th European Workshop on Computational Geometry CG’98 (Barcelona)

    2001 doi

  35. [42]

    Fox and X

    J. Fox and X. He. Independent sets in hypergraphs with a forbidden link. Proc. Lond. Math. Soc. (3) , 123(4):384–409, 2021. doi: 10.1112/plms.12400

    2021 doi

  36. [43]

    Fox and X

    J. Fox and X. He. Ramsey numbers of sparse digraphs. Israel J. Math. , pages 1–48,

  37. [44]

    Fox and R

    J. Fox and R. Li. On edge-ordered Ramsey numbers. Random Structures Algorithms, 57(4):1174–1204, 2020. doi: 10.1002/rsa.20954

    2020 doi

  38. [45]

    J. Fox, J. Pach, B. Sudakov, and A. Suk. Erd˝ os–Szekeres-type theorems for monotone paths and convex bodies. Proc. Lond. Math. Soc. (3) , 105(5):953–982, 2012. doi: 10.1112/plms/pds018

    2012 doi

  39. [46]

    Geneson, A

    J. Geneson, A. Holmes, X. Liu, D. Neidinger, Y. Pehova, and I. Wass. Ramsey numbers of ordered graphs under graph operations. http://arXiv.org/abs/1902.00259, 2019

    1902 arXiv

  40. [47]

    Gerbner, A

    D. Gerbner, A. Methuku, D. T. Nagy, D. P´ alv¨ olgyi, G. Tardos, and M. Vizer. Edge ordered Tur´ an problems.Acta Math. Univ. Comenian. (N.S.) , 88(3):717–722, 2019

    2019

  41. [48]

    Gir˜ ao, B

    A. Gir˜ ao, B. Janzer, and O. Janzer. Ordered Ramsey numbers of powers of paths. http://arXiv.org/abs/2401.02360, 2024

    2024 arXiv

  42. [49]

    Gishboliner, Z

    L. Gishboliner, Z. Jin, and B. Sudakov. Ramsey problems for monotone paths in graphs and hypergraphs. Combinatorica, 44(3):509–529, 2024. doi: 10.1007/s00493-024-00082-7

    2024 doi

  43. [50]

    Grinerov´ a

    K. Grinerov´ a. Multi-colored ordered ramsey numbers, 2024. Bachelor’s thesis

    2024

  44. [51]

    Hubiˇ cka and J

    J. Hubiˇ cka and J. Neˇ setˇ ril. All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms). Adv. Math., 356:106791, 89, 2019. doi: 10.1016/j.aim.2019. 106791

    2019 doi

  45. [52]

    Ishigami

    Y. Ishigami. Linear Ramsey numbers for bounded-degree hypergrahps. Electron. Notes Discret. Math., 29:47 – 51, 2007. doi: https://doi.org/10.1016/j.endm.2007.07.009

    2007 doi

  46. [53]

    Janzer, O

    B. Janzer, O. Janzer, V. Magnan, and A. Methuku. Tight General Bounds for the Extremal Numbers of 0–1 Matrices. Int. Math. Res. Not. IMRN , 2024(15):11455–11463,

    2024

  47. [54]

    K´ arolyi, J

    Gy. K´ arolyi, J. Pach, and G. T´ oth. Ramsey-type results for geometric graphs. I.Discrete Comput. Geom., 18(3):247–255, 1997. doi: 10.1007/PL00009317. ACM Symposium on Computational Geometry (Philadelphia, PA, 1996)

    1997 doi

  48. [55]

    K´ arolyi, J

    Gy. K´ arolyi, J. Pach, G. T´ oth, and P. Valtr. Ramsey-type results for geometric graphs. II. Discrete Comput. Geom. , 20(3):375–388, 1998. doi: 10.1007/PL00009391. ACM Symposium on Computational Geometry (Nice, 1997)

    1998 doi

  49. [56]

    J. H. Kim. The Ramsey number R(3,t ) has order of magnitude t2/ logt. Random Structures Algorithms, 7(3):173–207, 1995. doi: 10.1002/rsa.3240070302

    1995 doi

  50. [57]

    doi: 10.1093/imrn/rnae129. 22

    doi

  51. [58]

    H. Lefmann. A note on Ramsey numbers. Studia Sci. Math. Hungar. , 22(1-4):445–446, 1987

    1987

  52. [59]

    K. G. Milans, D. Stolee, and D. B. West. Ordered Ramsey theory and track representa- tions of graphs. J. Comb., 6(4):445–456, 2015. doi: 10.4310/JOC.2015.v6.n4.a3

    2015 doi

  53. [60]

    H. Miyata. On combinatorial properties of points and polynomial curves. http: //arxiv.org/abs/1703.04963, 2017

    2017 arXiv

  54. [61]

    Kleitman and L

    D. Kleitman and L. Pachter. Finding convex sets among points in the plane. Discrete Comput. Geom., 19(3, Special Issue):405–410, 1998. doi: 10.1007/PL00009358. Dedicated to the memory of Paul Erd˝ os

    1998 doi

  55. [62]

    Moshkovitz and A

    G. Moshkovitz and A. Shapira. Ramsey theory, integer partitions and a new proof of the Erd˝ os–Szekeres theorem.Adv. Math., 262:1107–1129, 2014. doi: 10.1016/j.aim.2014. 06.008

    2014 doi

  56. [63]

    D. Mubayi. Variants of the Erd˝ os–Szekeres and Erd˝ os-Hajnal Ramsey problems.Euro- pean J. Combin., 62:197–205, 2017. doi: 10.1016/j.ejc.2016.12.007

    2017 doi

  57. [65]

    Nassajian Mojarrad and G

    H. Nassajian Mojarrad and G. Vlachos. An improved upper bound for the Erd˝ os– Szekeres conjecture. Discrete Comput. Geom. , 56(1):165–180, 2016. doi: 10.1007/ s00454-016-9791-5

    2016

  58. [66]

    Mubayi and A

    D. Mubayi and A. Suk. Ramsey numbers of cliques versus monotone paths. European J. Combin., 118:Paper No. 103922, 7, 2024. doi: 10.1016/j.ejc.2024.103922

    2024

  59. [67]

    Nagle, S

    B. Nagle, S. Olsen, V. R¨ odl, and M. Schacht. On the Ramsey number of sparse 3-graphs. Graphs Combin., 24(3):205–228, 2008. doi: 10.1007/s00373-008-0784-x

    2008 doi

  60. [68]

    Neidinger and D

    D. Neidinger and D. B. West. Ramsey numbers of interval 2-chromatic ordered graphs. Graphs Combin., 35(5):1065–1076, 2019. doi: 10.1007/s00373-019-02057-8

    2019 doi

  61. [69]

    Mubayi and A

    D. Mubayi and A. Suk. A survey of hypergraph Ramsey problems. In Discrete mathematics and applications , volume 165 of Springer Optim. Appl. , pages 405–428. Springer, Cham, 2020. doi: 10.1007/978-3-030-55857- \ 16

    2020 doi

  62. [70]

    Pach and G

    J. Pach and G. Tardos. Forbidden paths and cycles in ordered graphs and matrices. Israel J. Math. , 155:359–380, 2006. doi: 10.1007/BF02773960

    2006 doi

  63. [71]

    F. P. Ramsey. On a Problem of Formal Logic. Proc. London Math. Soc. (2) , 30(4): 264–286, 1929. doi: 10.1112/plms/s2-30.1.264

    1929 doi

  64. [72]

    D. Rohatgi. Off-diagonal ordered Ramsey numbers of matchings. Electron. J. Combin., 26(2):Paper No. 2.21, 18, 2019

    2019

  65. [73]

    Norin and Y

    S. Norin and Y. Yuditsky. Erd˝ os–Szekeres without induction.Discrete Comput. Geom., 55(4), 2016. doi: 10.1007/s00454-016-9778-2. 23

    2016 doi

  66. [74]

    W. Sawin. An improved lower bound for multicolor Ramsey numbers and a problem of Erd˝ os.J. Combin. Theory Ser. A , 188:Paper No. 105579, 11, 2022. doi: 10.1016/j.jcta. 2021.105579

    2022

  67. [75]

    J. Spencer. Ramsey’s theorem—a new lower bound. J. Combinatorial Theory Ser. A , 18:108–115, 1975. doi: 10.1016/0097-3165(75)90071-0

    1975 doi

  68. [76]

    A. Suk. On the Erd˝ os–Szekeres convex polygon problem.J. Amer. Math. Soc. , 30(4): 1047–1053, 2017. doi: 10.1090/jams/869

    2017 doi

  69. [77]

    A. Sah. Diagonal Ramsey via effective quasirandomness. Duke Math. J., 172(3):545–567,

  70. [78]

    T´ oth and P

    G. T´ oth and P. Valtr. Note on the Erd˝ os–Szekeres theorem.Discrete Comput. Geom., 19(3, Special Issue):457–459, 1998. doi: 10.1007/PL00009363. Dedicated to the memory of Paul Erd˝ os

    1998 doi

  71. [79]

    Wigderson

    Y. Wigderson. An improved lower bound on multicolor Ramsey numbers. Proc. Amer. Math. Soc., 149(6):2371–2374, 2021. doi: 10.1090/proc/15447

    2021 doi

  72. [80]

    G. Ziegler. Higher Bruhat orders and cyclic hyperplane arrangements. Topology, 32(2): 259–279, 1993. doi: 10.1016/0040-9383(93)90019-R. 24

    1993 doi

  73. [82]

    Szekeres and L

    G. Szekeres and L. Peters. Computer solution to the 17-point Erd˝ os–Szekeres problem. ANZIAM J., 48(2):151–164, 2006. doi: 10.1017/S144618110000300X

    2006 doi

  74. [1947]

    doi: 10.1090/S0002-9904-1947-08785-1

    1947 doi

  75. [2008]

    doi: 10.1016/j.jctb.2007.08.008

    2007 doi

  76. [2023]

    doi: 10.1215/00127094-2022-0048

    2022 doi

  77. [2024]

    doi: 10.1007/s11856-024-2624-y

    doi

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.