{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:18:04Z","timestamp":1758824284094},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"06","license":[{"start":{"date-parts":[[2019,7,17]],"date-time":"2019-07-17T00:00:00Z","timestamp":1563321600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2019,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>For given graphs <jats:italic>G<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026, <jats:italic>G<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>, the size-Ramsey number <jats:inline-formula>\n<jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548319000221_inline1\" xlink:type=\"simple\" \/><jats:tex-math>\n$\\hat R({G_1}, \\ldots ,{G_k})$\n<\/jats:tex-math><\/jats:alternatives>\n<\/jats:inline-formula> is the smallest integer <jats:italic>m<\/jats:italic> for which there exists a graph <jats:italic>H<\/jats:italic> on <jats:italic>m<\/jats:italic> edges such that in every <jats:italic>k<\/jats:italic>-edge colouring of <jats:italic>H<\/jats:italic> with colours 1,\u2026,<jats:italic>k<\/jats:italic>, <jats:italic>H<\/jats:italic> contains a monochromatic copy of <jats:italic>G<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> of colour <jats:italic>i<\/jats:italic> for some 1 \u2264 <jats:italic>i<\/jats:italic> \u2264 <jats:italic>k<\/jats:italic>. We denote <jats:inline-formula>\n<jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548319000221_inline2\" xlink:type=\"simple\" \/><jats:tex-math>\n$\\hat R({G_1}, \\ldots ,{G_k})$\n<\/jats:tex-math><\/jats:alternatives>\n<\/jats:inline-formula> by <jats:inline-formula>\n<jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548319000221_inline3\" xlink:type=\"simple\" \/><jats:tex-math>\n${\\hat R_k}(G)$\n<\/jats:tex-math><\/jats:alternatives>\n<\/jats:inline-formula> when <jats:italic>G<\/jats:italic><jats:sub>1<\/jats:sub> = \u22ef = <jats:italic>G<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> = <jats:italic>G<\/jats:italic>.<\/jats:p><jats:p>Haxell, Kohayakawa and \u0141uczak showed that the size-Ramsey number of a cycle <jats:italic>C<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> is linear in <jats:italic>n<\/jats:italic>, <jats:inline-formula>\n<jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548319000221_inline4\" xlink:type=\"simple\" \/><jats:tex-math>\n${\\hat R_k}({C_n}) \\le {c_k}n$\n<\/jats:tex-math><\/jats:alternatives>\n<\/jats:inline-formula> for some constant <jats:italic>c<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>. Their proof, however, is based on Szemer\u00e9di\u2019s regularity lemma so no specific constant <jats:italic>c<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> is known.<\/jats:p><jats:p>In this paper, we give various upper bounds for the size-Ramsey numbers of cycles. We provide an alternative proof of <jats:inline-formula>\n<jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548319000221_inline5\" xlink:type=\"simple\" \/><jats:tex-math>\n${\\hat R_k}({C_n}) \\le {c_k}n$\n<\/jats:tex-math><\/jats:alternatives>\n<\/jats:inline-formula>, avoiding use of the regularity lemma, where <jats:italic>c<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> is exponential and doubly exponential in <jats:italic>k<\/jats:italic>, when <jats:italic>n<\/jats:italic> is even and odd, respectively. In particular, we show that for sufficiently large <jats:italic>n<\/jats:italic> we have <jats:inline-formula>\n<jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548319000221_inline6\" xlink:type=\"simple\" \/><jats:tex-math>\n${\\hat R_2}({C_n}) \\le {10^5} \\times cn$\n<\/jats:tex-math><\/jats:alternatives>\n<\/jats:inline-formula>, where <jats:italic>c<\/jats:italic> = 6.5 if <jats:italic>n<\/jats:italic> is even and <jats:italic>c<\/jats:italic> = 1989 otherwise.<\/jats:p>","DOI":"10.1017\/s0963548319000221","type":"journal-article","created":{"date-parts":[[2019,7,17]],"date-time":"2019-07-17T00:38:15Z","timestamp":1563323895000},"page":"871-880","source":"Crossref","is-referenced-by-count":13,"title":["On the Size-Ramsey Number of Cycles"],"prefix":"10.1017","volume":"28","author":[{"given":"R.","family":"Javadi","sequence":"first","affiliation":[]},{"given":"F.","family":"Khoeini","sequence":"additional","affiliation":[]},{"given":"G. R.","family":"Omidi","sequence":"additional","affiliation":[]},{"given":"A.","family":"Pokrovskiy","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2019,7,17]]},"reference":[{"key":"S0963548319000221_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02018930"},{"key":"S0963548319000221_ref10","first-page":"463","volume":"2","author":"Erd\u0151s","year":"1935","journal-title":"Compositio Math"},{"key":"S0963548319000221_ref7","doi-asserted-by":"publisher","DOI":"10.1137\/16M1069717"},{"key":"S0963548319000221_ref6","doi-asserted-by":"publisher","DOI":"10.1017\/S096354831400056X"},{"key":"S0963548319000221_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781316106853.003"},{"key":"S0963548319000221_ref4","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511814068"},{"key":"S0963548319000221_ref16","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s2-30.1.264"},{"key":"S0963548319000221_ref3","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190070115"},{"key":"S0963548319000221_ref15","volume":"1","author":"Radziszowski","year":"1994","journal-title":"Electron. 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