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Infinite monochromatic paths and a theorem of Erdos-Hajnal-Rado [article]

Shimon Garti and Menachem Magidor and Saharon Shelah
<span title="2020-03-29">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove that if μ> cf(μ)=ω and 2^μ=μ^+ then μ^+μμ^+ ℵ_2μ μ.  ...  In order to prove our result we replace the Erdős-Rado theorem by a statement about monochromatic paths in complete graphs. We shall use a theorem of Todorčević from [15] , see also [12] .  ...  This theorem is essential for proving the negative polarized relation µ + µ µ + ω 2 µ µ . The employment of the Erdős-Rado theorem invites for a natural question.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1907.03254v2">arXiv:1907.03254v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6k2s3qjbdbcozp43vxvpd4gd2y">fatcat:6k2s3qjbdbcozp43vxvpd4gd2y</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200408071700/https://arxiv.org/pdf/1907.03254v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1907.03254v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Infinite Monochromatic Paths and a Theorem of Erdős-Hajnal-Rado

Shimon Garti, Menachem Magidor, Saharon Shelah
<span title="2020-04-17">2020</span> <i title="The Electronic Journal of Combinatorics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v5dyak6ulffqfara7hmuchh24a" style="color: black;">Electronic Journal of Combinatorics</a> </i> &nbsp;
We prove that if $\mu$ is a singular cardinal with countable cofinality and $2^\mu=\mu^+$ then $\binom{\mu^+}{\mu}\nrightarrow\binom{\mu^+\ \aleph_2}{\mu\ \mu}$.  ...  Acknowledgements The first and the third authors gratefully acknowledge the support of the ERC, grant no. 338821. This is publication 1165 of the third author.  ...  In order to prove our result we replace the Erdős-Rado theorem by a statement about monochromatic paths in complete graphs. We shall use a theorem of Todorčević from [15] , see also [12] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/8849">doi:10.37236/8849</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gtvjlzemcnd4dork3p3jn3mfrq">fatcat:gtvjlzemcnd4dork3p3jn3mfrq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200508101034/https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i2p8/8065" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ef/c9/efc98180059455587b9f184479da0a7b44f0b459.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/8849"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Off-diagonal hypergraph Ramsey numbers [article]

Dhruv Mubayi, Andrew Suk
<span title="2015-05-29">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
determining the tower growth rate of r_k(n,n), which is a notorious conjecture of Erdos, Hajnal and Rado from 1965 that remains open.  ...  The question of determining the tower growth rate of r_k(s,n) for all s > k+1 was posed by Erdos and Hajnal in 1972. (2) We show that determining the tower growth rate of r_k(P_k+1, n) is equivalent to  ...  of Erdős and Hajnal.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1505.05767v2">arXiv:1505.05767v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/o6cb7b42f5c4jf3kkymuxiviz4">fatcat:o6cb7b42f5c4jf3kkymuxiviz4</a> </span>
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Vertex coverings by monochromatic cycles and trees

P Erdős, A Gyárfás, L Pyber
<span title="">1991</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g6u5fful5vcr3a7gppc6y47el4" style="color: black;">Journal of combinatorial theory. Series B (Print)</a> </i> &nbsp;
If the edges of a finite complete graph K are colored with r colors then the vertex set of K can be covered by at most cr2 log r vertex disjoint monochromatic cycles.  ...  ACKNOWLEDGMENTS The authors are grateful to referees whose remarks led to reorganizing the original version of this paper.  ...  The case r = 2 in Conjecture 2 is equivalent with the fact that for any graph G, either G or its complement is connected, an old remark of Erdos and Rado. The case r = 3 is settled by THEOREM 2.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0095-8956(91)90007-7">doi:10.1016/0095-8956(91)90007-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dkyocj5f3jfvtdkqngr6e72nay">fatcat:dkyocj5f3jfvtdkqngr6e72nay</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190415015804/https://core.ac.uk/download/pdf/82587629.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c2/b3/c2b3379811dbe565f6952083b7e72ba1ff3a369d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0095-8956(91)90007-7"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Partitioning by Monochromatic Trees

P.E. Haxell, Y. Kohayakawa
<span title="">1996</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g6u5fful5vcr3a7gppc6y47el4" style="color: black;">Journal of combinatorial theory. Series B (Print)</a> </i> &nbsp;
This comes close to proving, for large n, a conjecture of Erdo s, Gya rfa s, and Pyber, which states that r&1 trees suffice for all n.  ...  Any r-edge-coloured n-vertex complete graph K n contains at most r monochromatic trees, all of different colours, whose vertex sets partition the vertex set of K n , provided n 3r 4 r!  ...  Rado [5] proved that an r-edge-coloured countably infinite complete graph may be partitioned into r monochromatic, possibly one-way infinite, paths, generalising a result of Erdo s, who had proved this  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jctb.1996.0065">doi:10.1006/jctb.1996.0065</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7ueyqnojcfbe7otdjjpejiqde4">fatcat:7ueyqnojcfbe7otdjjpejiqde4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170930190941/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/94a/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDA5NTg5NTY5NjkwMDY1OQ%3D%3D.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/5e/99/5e99ce999676378878e3d7206da05dffef899c11.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jctb.1996.0065"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Recent developments in graph Ramsey theory [article]

David Conlon, Jacob Fox, Benny Sudakov
<span title="2015-05-10">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K_N contains a monochromatic copy of H.  ...  Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics.  ...  The authors would like to thank the anonymous referee for a number of useful comments.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1501.02474v3">arXiv:1501.02474v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/q3qcowfhgjbp5j36oetad6qdnq">fatcat:q3qcowfhgjbp5j36oetad6qdnq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200913102852/https://arxiv.org/pdf/1501.02474v3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/6e/06/6e0633ff2e01ec64d58ee15a70bc959e6cc83099.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1501.02474v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Open problems of Paul Erd�s in graph theory

F. R. K. Chung
<span title="">1997</span> <i title="Wiley"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ukzsjb6a6zhyxnjl6nb2mmjc6m" style="color: black;">Journal of Graph Theory</a> </i> &nbsp;
There is a huge literature of almost 1500 papers written by Erdős and his (more than 460) collaborators.  ...  of Paul Erdős", Volumes I and II [81] .  ...  During the preparation of this paper, many people have given numerous valuable comments and suggestions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/(sici)1097-0118(199705)25:1&lt;3::aid-jgt1&gt;3.0.co;2-r">doi:10.1002/(sici)1097-0118(199705)25:1&lt;3::aid-jgt1&gt;3.0.co;2-r</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ja46si6w75gvzh2ksrkvblnznm">fatcat:ja46si6w75gvzh2ksrkvblnznm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808150123/http://perso.ens-lyon.fr:80/eric.thierry/Graphes2010/erdos-conjectures.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/95/81/95818d55fe5e4668ba66b0238dff86e15b47a14f.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/(sici)1097-0118(199705)25:1&lt;3::aid-jgt1&gt;3.0.co;2-r"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> wiley.com </button> </a>

Monochromatic and Heterochromatic Subgraphs in Edge-Colored Graphs - A Survey

Mikio Kano, Xueliang Li
<span title="">2008</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yxooi3wjmbgqtbp7l4evjeuj4i" style="color: black;">Graphs and Combinatorics</a> </i> &nbsp;
We have to point out that there are a lot of results of Ramsey type problem on monochromatic and heterochromatic subgraphs.  ...  complexity of these partition problems; some kinds of large monochromatic and heterochromatic subgraphs.  ...  The case r = 2 in Conjecture 5 is equivalent to the fact that for any graph G, either G or its complement is connected, an old remark of Erdös and Rado.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00373-008-0789-5">doi:10.1007/s00373-008-0789-5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/a4j64ccspze2td7cv3f247kf7u">fatcat:a4j64ccspze2td7cv3f247kf7u</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809025836/http://gorogoro.cis.ibaraki.ac.jp/web/papers/kano2008-b.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/60/5f/605fc76becbf10cd7ce0ec375d18fca86553b4ca.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00373-008-0789-5"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Embedding graphs into larger graphs: results, methods, and problems [article]

Miklós Simonovits, Endre Szemerédi
<span title="2019-12-04">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Extremal Graph Theory is a very deep and wide area of modern combinatorics.  ...  It is very fast developing, and in this long but relatively short survey we select some of those results which either we feel very important in this field or which are new breakthrough results, or which  ...  We thank above all to József Balogh and András Gyárfás, and also to Zoltán Füredi, János Pach, Jan Hladký, Zoltán L.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1912.02068v1">arXiv:1912.02068v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2rgpm6wbmvafta74cmcbbxhcs4">fatcat:2rgpm6wbmvafta74cmcbbxhcs4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200906025654/https://arxiv.org/pdf/1912.02068v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/33/aa/33aa1aec19a5ffc2f071d07c2b4902ec6862fc3a.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1912.02068v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Ramsey Theory on Infinite Structures and the Method of Strong Coding Trees [article]

Natasha Dobrinen
<span title="2020-09-08">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Ramsey theory of copies of the Rado graph.  ...  Using forcing as a tool for finite searches has allowed the development of Ramsey theory on such trees, leading to solutions for finite big Ramsey degrees of Henson graphs as well as infinite dimensional  ...  Erdős, Hajnal, and Pósa launched the investigation of finite big Ramsey degrees of the Rado graph in 1975, when they showed that there is a 2-coloring of edges such that every subcopy of the Rado graph  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1909.05985v2">arXiv:1909.05985v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/66ixjzmsqrgu7eluzq5rzxz77a">fatcat:66ixjzmsqrgu7eluzq5rzxz77a</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200918205848/https://arxiv.org/pdf/1909.05985v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/aa/89/aa895c8bd7641e42e3f928a98d925272240e220e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1909.05985v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Multicolor Ramsey numbers for triple systems [article]

Maria Axenovich, Andras Gyarfas, Hong Liu, Dhruv Mubayi
<span title="2013-02-21">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We give a proof for cliques of all sizes, s>r, using a slight modification of the celebrated stepping-up lemma of Erdős and Hajnal.  ...  Erdős, Hajnal and Rado gave bounds for large cliques K_s^r with s> s_0(r), showing its correct exponential tower growth.  ...  Acknowledgments Thanks to Zoli Füredi and Roman Glebov for conversations on the subject of this paper, and Stefan Walzer for improving the lower bound in Theorem 6.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1302.5304v1">arXiv:1302.5304v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3gihbc3webai5koy7sjwjdu5r4">fatcat:3gihbc3webai5koy7sjwjdu5r4</a> </span>
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Multicolor Ramsey numbers for triple systems

Maria Axenovich, András Gyárfás, Hong Liu, Dhruv Mubayi
<span title="">2014</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
We give a proof for cliques of all sizes, s > r, using a slight modification of the celebrated stepping-up lemma of Erdős and Hajnal.  ...  Erdős, Hajnal and Rado gave bounds for large cliques K r s with s ≥ s 0 (r), showing its correct exponential tower growth.  ...  Acknowledgments Thanks to Zoli Füredi and Roman Glebov for conversations on the subject of this paper, and Stefan Walzer for improving the lower bound in Theorem 6.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2014.01.004">doi:10.1016/j.disc.2014.01.004</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hw2go52cuzbz7b5duf7cnfaxom">fatcat:hw2go52cuzbz7b5duf7cnfaxom</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170810042130/http://www.math.kit.edu/iag6/~axenovich/media/hyp-ram-feb20.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/bf/27/bf27c2f3153749f793fa1d4f0d93d5750916e347.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2014.01.004"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Reflections on Paul Erdős on His Birth Centenary

Krishnaswami Alladi, Steven Krantz, László Lovász, Vera T. Sós, Ronald L Graham, Joel Spencer, Jean-Pierre Kahane, Melvyn B. Nathanson
<span title="2015-02-01">2015</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/o5hwtvfzwza2tdpaa2xm3sqpqe" style="color: black;">Notices of the American Mathematical Society</a> </i> &nbsp;
This is Part I of a two-part feature on Paul Erdős following his centennial.  ...  There are eleven articles by leading experts who have reflected on the remarkable life, contributions, and influence of this towering figure of twentieth century mathematics.  ...  1 László Alpár , Pál Erdős (1913Erdős ( -1996, János Erőds, , Ervin Feldheim , Géza Grünwald (1913Grünwald ( -1944, Tibor Grünwald (Gallai) , Eszter Klein (1910Klein ( -1975, Dezső Lázár , György Szekeres  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/noti1211">doi:10.1090/noti1211</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jc6kwszplnb5fn24gu2sf6cwwa">fatcat:jc6kwszplnb5fn24gu2sf6cwwa</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180605052116/http://www.ams.org/notices/201502/rnoti-p121.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/db/8b/db8b4499e60c6303b614bcd14772d982c389b412.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/noti1211"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Decompositions of edge-coloured infinite complete graphs into monochromatic paths II [article]

Daniel T. Soukup
<span title="2016-03-16">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove that the vertices of every finite-edge coloured infinite complete graph can be partitioned into disjoint monochromatic paths of different colours. This answers a question of R.  ...  Erdős proved that every 2-edge coloured complete graph on the natural numbers can be vertex decomposed into two monochromatic paths of different colour. This result was extended by R.  ...  Rado asked if the path decomposition result of Erdős concerning the complete graph on N extends to uncountable complete graphs of arbitrary size.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1507.06187v2">arXiv:1507.06187v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/voz2mkrn5jasph5cwgtvmqmpvu">fatcat:voz2mkrn5jasph5cwgtvmqmpvu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200910063200/https://arxiv.org/pdf/1507.06187v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/74/43/7443de02415e8e2f94d408dcff3ba680f63b47c9.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1507.06187v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Borel sets of Rado graphs and Ramsey's Theorem [article]

Natasha Dobrinen
<span title="2020-07-01">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this article, we prove an analogue of the Galvin-Prikry theorem for the Rado graph.  ...  Any such infinite dimensional Ramsey theorem is subject to constraints following from the 2006 work of Laflamme, Sauer, and Vuksanovic.  ...  Here, we mention a theorem of Erdős and Rado which will be used in the proof of the main theorem of this section.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1904.00266v4">arXiv:1904.00266v4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/au32ivuv3jhrzfgekgyckjll2u">fatcat:au32ivuv3jhrzfgekgyckjll2u</a> </span>
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