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Lower bounds on geometric Ramsey functions
[article]

2014
*
arXiv
*
pre-print

Together with a recent work of Bárány, Matoušek, and Pór, our results imply a tower

arXiv:1307.5157v3
fatcat:3wwtt5pyybc23l5zhtg4mjjtty
*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). ...##
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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

2004
*
Electronic Journal of Combinatorics
*

Relations of

doi:10.37236/34
fatcat:gxrfo23hzzewjg7rez76d4xx4i
*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. ...##
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A Ramsey-Type Result for Geometric ℓ-Hypergraphs
[chapter]

2013
*
Lecture Notes in Computer Science
*

This is in contrast to the fact that the upper and

doi:10.1007/978-3-319-03841-4_32
fatcat:pzficny3lndz3gabiblldfcerq
*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). ...##
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A Ramsey-type result for geometric l-hypergraphs
[article]

2014
*
arXiv
*
pre-print

This is in contrast to the fact that the upper and

arXiv:1305.5227v3
fatcat:gsi7rnactng6novxzvtvsq2oam
*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). ...##
###
Ramsey Numbers of Ordered Graphs

2020
*
Electronic Journal of Combinatorics
*

For a few special classes of ordered paths, stars or matchings, we give asymptotically tight

doi:10.37236/7816
fatcat:57evli25fzb4hm3owvapqgzv74
*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 ...##
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Ramsey numbers of ordered graphs
[article]

2019
*
arXiv
*
pre-print

For a few special classes of ordered paths, stars or matchings, we give asymptotically tight

arXiv:1310.7208v5
fatcat:ok5oidqvojfbxphlaqknujutpq
*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 ...##
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Coloring arcs of convex sets

2000
*
Discrete Mathematics
*

We will give

doi:10.1016/s0012-365x(99)00403-3
fatcat:2ygvqs6725cyhdx6khz7awtwza
*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. ...##
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Configurations in projective planes and quadrilateral-star Ramsey numbers

2008
*
Discrete Mathematics
*

As the main result,

doi:10.1016/j.disc.2007.07.022
fatcat:wd4fypwtdjgcloeguxo3uhlmqy
*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. ...##
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Higher-order Erdos--Szekeres theorems
[article]

2012
*
arXiv
*
pre-print

We obtain an $\Omega(\log^{(k-1)}N)$

arXiv:1111.3824v3
fatcat:e2m3u5dnfjffbfx44c2r6ggqdi
*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. ...##
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Ramsey–Turán theory

2001
*
Discrete Mathematics
*

For every L 1 ; : : : ; L r there exists a

doi:10.1016/s0012-365x(00)00214-4
fatcat:2nfhfsiozna5lkc2flkpubbkva
*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. ...##
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Ramsey Theory in the Work of Paul Erdős
[chapter]

1997
*
Algorithms and Combinatorics
*

But perhaps

doi:10.1007/978-3-642-60406-5_16
fatcat:cqkc3inzjfe4lbu2jj4oem2fv4
*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*. ...##
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Ramsey Theory in the Work of Paul Erdős
[chapter]

2013
*
The Mathematics of Paul Erdős II
*

But perhaps

doi:10.1007/978-1-4614-7254-4_13
fatcat:vqvyhfhj3ngr7c3vokrsmrt2ai
*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*. ...##
###
A note on Sidon-Ramsey numbers
[article]

2021
*
arXiv
*
pre-print

Given a positive integer k, the Sidon-

arXiv:2111.08076v1
fatcat:mge2enabgjggxnaprkrecghkcq
*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. ...##
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Applications of the Canonical Ramsey Theorem to Geometry
[article]

2013
*
arXiv
*
pre-print

All of our proofs use variants of the canonical

arXiv:1302.5334v1
fatcat:7alp725zwbgbhlxcxnbhgxwcsy
*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. ...
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