Ramsey-type numbers involving graphs and hypergraphs with large girth
Erdős asked if, for every pair of positive integers g and k , there exists a graph H having girth ( H ) = k and the property that every r -colouring of the edges of H yields a monochromatic cycle C k . The existence of such graphs H was confirmed by the third author and Ruciński. We consider the rel...
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| Veröffentlicht in: | Combinatorics, probability & computing probability & computing, 2021-09, Vol.30 (5), p.722-740 |
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| container_title | Combinatorics, probability & computing |
| container_volume | 30 |
| creator | Hàn, Hiêp Retter, Troy Rödl, Vojtêch Schacht, Mathias |
| description | Erdős asked if, for every pair of positive integers
g
and
k
, there exists a graph
H
having girth (
H
) =
k
and the property that every
r
-colouring of the edges of
H
yields a monochromatic cycle
C
k
. The existence of such graphs
H
was confirmed by the third author and Ruciński.
We consider the related numerical problem of estimating the order of the smallest graph
H
with this property for given integers
r
and
k
. We show that there exists a graph
H
on
R
10
k
2
;
k
15
k
3
vertices (where
R
=
R
(
C
k
;
r
) is the
r
-colour Ramsey number for the cycle
C
k
) having girth (
H
) =
k
and the Ramsey property that every
r
-colouring of the edges of
H
yields a monochromatic
C
k
Two related numerical problems regarding arithmetic progressions in subsets of the integers and cliques in graphs are also considered. |
| doi_str_mv | 10.1017/S0963548320000383 |
| format | Article |
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g
and
k
, there exists a graph
H
having girth (
H
) =
k
and the property that every
r
-colouring of the edges of
H
yields a monochromatic cycle
C
k
. The existence of such graphs
H
was confirmed by the third author and Ruciński.
We consider the related numerical problem of estimating the order of the smallest graph
H
with this property for given integers
r
and
k
. We show that there exists a graph
H
on
R
10
k
2
;
k
15
k
3
vertices (where
R
=
R
(
C
k
;
r
) is the
r
-colour Ramsey number for the cycle
C
k
) having girth (
H
) =
k
and the Ramsey property that every
r
-colouring of the edges of
H
yields a monochromatic
C
k
Two related numerical problems regarding arithmetic progressions in subsets of the integers and cliques in graphs are also considered.</description><identifier>ISSN: 0963-5483</identifier><identifier>EISSN: 1469-2163</identifier><identifier>DOI: 10.1017/S0963548320000383</identifier><language>eng</language><publisher>Cambridge: Cambridge University Press</publisher><subject>Apexes ; Coloring ; Graph theory ; Graphs ; Integers ; Numbers ; Progressions</subject><ispartof>Combinatorics, probability & computing, 2021-09, Vol.30 (5), p.722-740</ispartof><rights>The Author(s), 2021. Published by Cambridge University Press</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c225t-457c050a7ebc38e093068a9c2be07c1b477445797784b91dd53b1ecc35ab89813</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27901,27902</link.rule.ids></links><search><creatorcontrib>Hàn, Hiêp</creatorcontrib><creatorcontrib>Retter, Troy</creatorcontrib><creatorcontrib>Rödl, Vojtêch</creatorcontrib><creatorcontrib>Schacht, Mathias</creatorcontrib><title>Ramsey-type numbers involving graphs and hypergraphs with large girth</title><title>Combinatorics, probability & computing</title><description>Erdős asked if, for every pair of positive integers
g
and
k
, there exists a graph
H
having girth (
H
) =
k
and the property that every
r
-colouring of the edges of
H
yields a monochromatic cycle
C
k
. The existence of such graphs
H
was confirmed by the third author and Ruciński.
We consider the related numerical problem of estimating the order of the smallest graph
H
with this property for given integers
r
and
k
. We show that there exists a graph
H
on
R
10
k
2
;
k
15
k
3
vertices (where
R
=
R
(
C
k
;
r
) is the
r
-colour Ramsey number for the cycle
C
k
) having girth (
H
) =
k
and the Ramsey property that every
r
-colouring of the edges of
H
yields a monochromatic
C
k
Two related numerical problems regarding arithmetic progressions in subsets of the integers and cliques in graphs are also considered.</description><subject>Apexes</subject><subject>Coloring</subject><subject>Graph theory</subject><subject>Graphs</subject><subject>Integers</subject><subject>Numbers</subject><subject>Progressions</subject><issn>0963-5483</issn><issn>1469-2163</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><sourceid>BENPR</sourceid><recordid>eNplkEtLxDAUhYMoWEd_gLuA62oezWspw-gIA4KPdUnSTNuhL5N2pP_elJmdd3O4nI9z4ABwj9EjRlg8fSLFKcskJSgelfQCJDjjKiWY00uQLHa6-NfgJoRDZBjjKAGbD90GN6fjPDjYTa1xPsC6O_bNse5KWHo9VAHqroBVJPz5_63HCjbalw6WtR-rW3C1101wd2ddge-Xzdd6m-7eX9_Wz7vUEsLGNGPCIoa0cMZS6ZCiiEutLDEOCYtNJkQWGSWEzIzCRcGowc5ayrSRSmK6Ag-n3MH3P5MLY37oJ9_FypwwIbiQEtFI4RNlfR-Cd_t88HWr_ZxjlC9r5f_Won-G6Fv3</recordid><startdate>202109</startdate><enddate>202109</enddate><creator>Hàn, Hiêp</creator><creator>Retter, Troy</creator><creator>Rödl, Vojtêch</creator><creator>Schacht, Mathias</creator><general>Cambridge University Press</general><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7SC</scope><scope>7XB</scope><scope>8AL</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>GNUQQ</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K7-</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0N</scope><scope>M7S</scope><scope>P5Z</scope><scope>P62</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>Q9U</scope></search><sort><creationdate>202109</creationdate><title>Ramsey-type numbers involving graphs and hypergraphs with large girth</title><author>Hàn, Hiêp ; Retter, Troy ; Rödl, Vojtêch ; Schacht, Mathias</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c225t-457c050a7ebc38e093068a9c2be07c1b477445797784b91dd53b1ecc35ab89813</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Apexes</topic><topic>Coloring</topic><topic>Graph theory</topic><topic>Graphs</topic><topic>Integers</topic><topic>Numbers</topic><topic>Progressions</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Hàn, Hiêp</creatorcontrib><creatorcontrib>Retter, Troy</creatorcontrib><creatorcontrib>Rödl, Vojtêch</creatorcontrib><creatorcontrib>Schacht, Mathias</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>Computing Database (Alumni Edition)</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>ProQuest Central Student</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>Computer Science Database</collection><collection>ProQuest Engineering Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Computing Database</collection><collection>Engineering Database</collection><collection>Advanced Technologies & Aerospace Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><collection>ProQuest Central Basic</collection><jtitle>Combinatorics, probability & computing</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Hàn, Hiêp</au><au>Retter, Troy</au><au>Rödl, Vojtêch</au><au>Schacht, Mathias</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Ramsey-type numbers involving graphs and hypergraphs with large girth</atitle><jtitle>Combinatorics, probability & computing</jtitle><date>2021-09</date><risdate>2021</risdate><volume>30</volume><issue>5</issue><spage>722</spage><epage>740</epage><pages>722-740</pages><issn>0963-5483</issn><eissn>1469-2163</eissn><abstract>Erdős asked if, for every pair of positive integers
g
and
k
, there exists a graph
H
having girth (
H
) =
k
and the property that every
r
-colouring of the edges of
H
yields a monochromatic cycle
C
k
. The existence of such graphs
H
was confirmed by the third author and Ruciński.
We consider the related numerical problem of estimating the order of the smallest graph
H
with this property for given integers
r
and
k
. We show that there exists a graph
H
on
R
10
k
2
;
k
15
k
3
vertices (where
R
=
R
(
C
k
;
r
) is the
r
-colour Ramsey number for the cycle
C
k
) having girth (
H
) =
k
and the Ramsey property that every
r
-colouring of the edges of
H
yields a monochromatic
C
k
Two related numerical problems regarding arithmetic progressions in subsets of the integers and cliques in graphs are also considered.</abstract><cop>Cambridge</cop><pub>Cambridge University Press</pub><doi>10.1017/S0963548320000383</doi><tpages>19</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0963-5483 |
| ispartof | Combinatorics, probability & computing, 2021-09, Vol.30 (5), p.722-740 |
| issn | 0963-5483 1469-2163 |
| language | eng |
| recordid | cdi_proquest_journals_2577678803 |
| source | Cambridge University Press Journals Complete |
| subjects | Apexes Coloring Graph theory Graphs Integers Numbers Progressions |
| title | Ramsey-type numbers involving graphs and hypergraphs with large girth |
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