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Lower bounds on geometric Ramsey functions [article]

Marek Eliáš, Jiří Matoušek, Edgardo Roldán-Pensado, Zuzana Safernová
2014 arXiv   pre-print
Together with a recent work of Bárány, Matoušek, and Pór, our results imply a tower function of Ω(n) of height d as a lower bound, matching an upper bound by Suk up to the constant in front of n.  ...  We also provide a natural geometric Ramsey-type theorem with a large Ramsey function. We call a point sequence P in R^d order-type homogeneous if all (d+1)-tuples in P have the same orientation.  ...  is bounded above by a tower one lower than the "combinatorial" Ramsey function R k (n).  ... 
arXiv:1307.5157v3 fatcat:3wwtt5pyybc23l5zhtg4mjjtty

Page 5483 of Mathematical Reviews Vol. , Issue 2002H [page]

2002 Mathematical Reviews  
The upper bound is obtained by finding an upper bound for a similar function P,(m,r) which is both bigger than R,(m,r) and easier to bound, for large m, by inducting on 1.  ...  They remark that by using a probabilistic argument different from the one used here, the upper bounds may be lowered by a factor of about /2.  ... 

Ramsey Theory Applications

Vera Rosta
2004 Electronic Journal of Combinatorics  
Relations of Ramsey-type theorems to various fields in mathematics are well documented in published books and monographs.  ...  There are many interesting applications of Ramsey theory, these include results in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and theoretical computer  ...  Section 9 mentions lower bounds for the Boolean function computation complexity with a Ramsey theoretical lemma developed for these.  ... 
doi:10.37236/34 fatcat:gxrfo23hzzewjg7rez76d4xx4i

A Ramsey-Type Result for Geometric ℓ-Hypergraphs [chapter]

Dhruv Mubayi, Andrew Suk
2013 Lecture Notes in Computer Science  
This is in contrast to the fact that the upper and lower bounds for the usual 3-uniform hypergraph Ramsey problem for two colors differ by one exponential in the tower.  ...  geometric graph on n vertices.  ...  Substituting in the lower bound on |S t |, we obtain the desired bound |S t+1 | ≥ N (t + 1)!q t+1 − (t + 1).  ... 
doi:10.1007/978-3-319-03841-4_32 fatcat:pzficny3lndz3gabiblldfcerq

A Ramsey-type result for geometric l-hypergraphs [article]

Dhruv Mubayi, Andrew Suk
2014 arXiv   pre-print
This is in contrast to the fact that the upper and lower bounds for the usual 3-uniform hypergraph Ramsey problem for two colors differ by one exponential in the tower.  ...  geometric graph on n vertices.  ...  Substituting in the lower bound on |S t |, we obtain the desired bound |S t+1 | ≥ N (t + 1)!q t+1 − (t + 1).  ... 
arXiv:1305.5227v3 fatcat:gsi7rnactng6novxzvtvsq2oam

Ramsey Numbers of Ordered Graphs

Martin Balko, Josef Cibulka, Karel Král, Jan Kynčl
2020 Electronic Journal of Combinatorics  
For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers.  ...  For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi, Pach, Tóth, and Valtr.  ...  Lower bound for ordered stars with interval chromatic number 3 We give a lower bound for ordered Ramsey numbers of ordered stars that have at least one edge incident to the central vertex from each side  ... 
doi:10.37236/7816 fatcat:57evli25fzb4hm3owvapqgzv74

Ramsey numbers of ordered graphs [article]

Martin Balko, Josef Cibulka, Karel Král, Jan Kynčl
2019 arXiv   pre-print
For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers.  ...  For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi, Pach, Tóth, and Valtr.  ...  Lower bound for ordered stars with interval chromatic number 3 We give a lower bound for ordered Ramsey numbers of ordered stars that have at least one edge incident to the central vertex from each side  ... 
arXiv:1310.7208v5 fatcat:ok5oidqvojfbxphlaqknujutpq

Coloring arcs of convex sets

Heiko Harborth, Hanno Lefmann
2000 Discrete Mathematics  
We will give lower and upper bounds for these geometric Ramsey numbers for certain paths and cycles and also some exact values.  ...  In this note we consider Ramsey-type problems on graphs whose vertices are represented by the vertices of a convex polygon in the Euclidean plane.  ...  For t even, the lower bound follows from the lower bound for t − 1.  ... 
doi:10.1016/s0012-365x(99)00403-3 fatcat:2ygvqs6725cyhdx6khz7awtwza

Configurations in projective planes and quadrilateral-star Ramsey numbers

E.L. Monte Carmelo
2008 Discrete Mathematics  
As the main result, lower bounds for the Ramsey numbers r(n) = r(C 4 ; K 1,n ) are derived from these geometric structures, which improve some bounds due to Parsons about 30 years ago, and also yield a  ...  Moreover, the constructions also imply a known result on C 4 − K 1,n bipartite Ramsey numbers.  ...  On the other hand, the lower bound is based on the Levi transformation (see details in [11] ), described as follows.  ... 
doi:10.1016/j.disc.2007.07.022 fatcat:wd4fypwtdjgcloeguxo3uhlmqy

Higher-order Erdos--Szekeres theorems [article]

Marek Elias, Jiri Matousek
2012 arXiv   pre-print
We obtain an $\Omega(\log^{(k-1)}N)$ lower bound ((k-1)-times iterated logarithm).  ...  For k=3, we construct a geometric example providing an $O(\log\log N)$ upper bound, tight up to a multiplicative constant.  ...  A Ramsey function with known doubly exponential growth seems to be rare in geometric Ramsey-type problems (a notable example is a result of Valtr [Val04] ). Order types.  ... 
arXiv:1111.3824v3 fatcat:e2m3u5dnfjffbfx44c2r6ggqdi

Ramsey–Turán theory

Miklós Simonovits, Vera T. Sós
2001 Discrete Mathematics  
For every L 1 ; : : : ; L r there exists a function f(n) = o(n) for which RT(n; L 1 ; : : : ; L r ; f(n)) = #(L 1 ; : : : ; L r )n 2 + o(n 2 ): See also Deÿnition 52 and Problem 9 on threshold functions  ...  Ramsey theory. • Some applications.  ...  This is why we need to put a lower bound Ä 0 on the number of classes.  ... 
doi:10.1016/s0012-365x(00)00214-4 fatcat:2nfhfsiozna5lkc2flkpubbkva

Ramsey Theory in the Work of Paul Erdős [chapter]

R. L. Graham, J. Nešetřil
1997 Algorithms and Combinatorics  
But perhaps one could say that Ramsey theory was created largely by him. This paper will attempt to demonstrate this claim.  ...  In Section 2, we consider the development based on Erd os' work related to bounds on various Ramsey functions.  ...  The best constructive l o wer bound for Ramsey numbers rn is due to Frankl and Wilson. This improves on an earlier construction of Frankl 46 who found a rst constructive superpolynomial lower bound.  ... 
doi:10.1007/978-3-642-60406-5_16 fatcat:cqkc3inzjfe4lbu2jj4oem2fv4

Ramsey Theory in the Work of Paul Erdős [chapter]

Ron L. Graham, Jaroslav Nešetřil
2013 The Mathematics of Paul Erdős II  
But perhaps one could say that Ramsey theory was created largely by him. This paper will attempt to demonstrate this claim.  ...  In Section 2, we consider the development based on Erd os' work related to bounds on various Ramsey functions.  ...  The best constructive l o wer bound for Ramsey numbers rn is due to Frankl and Wilson. This improves on an earlier construction of Frankl 46 who found a rst constructive superpolynomial lower bound.  ... 
doi:10.1007/978-1-4614-7254-4_13 fatcat:vqvyhfhj3ngr7c3vokrsmrt2ai

A note on Sidon-Ramsey numbers [article]

Manuel A. Espinosa-García, Amanda Montejano, Edgardo Roldán-Pensado, J. David Suárez
2021 arXiv   pre-print
Given a positive integer k, the Sidon-Ramsey number SR(k) is defined as the minimum n such that in every partition of [1,n] into k parts there is a part containing two pairs of numbers with the same sum  ...  As mentioned above, it is possible to use bounds on density problems to give bounds on Ramsey problems.  ...  The O(k 5/3 ) term in the lower bound depends on the upper bounds for the prime gap problem [BHP01] . Finally, there is also a Ramsey version for the box case.  ... 
arXiv:2111.08076v1 fatcat:mge2enabgjggxnaprkrecghkcq

Applications of the Canonical Ramsey Theorem to Geometry [article]

William Gasarch, Sam Zbarsky
2013 arXiv   pre-print
All of our proofs use variants of the canonical Ramsey theorem and some geometric lemmas.  ...  We would like to thank David Conlon and Jacob Fox for thoughtful discussions, many references and observations, encouragement, and advice on this paper.  ...  Lower Bounds on h 3,2 and h 3,3 For the problem of h 2,d we used (1) upper bounds on the asymmetric weak canonical Ramsey theorem and (2) a geometric lemma.  ... 
arXiv:1302.5334v1 fatcat:7alp725zwbgbhlxcxnbhgxwcsy
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