{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T06:06:20Z","timestamp":1757311580811,"version":"3.37.3"},"reference-count":10,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,5,31]],"date-time":"2023-05-31T00:00:00Z","timestamp":1685491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,5,31]],"date-time":"2023-05-31T00:00:00Z","timestamp":1685491200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["FKZ AX 93\/2-1"],"award-info":[{"award-number":["FKZ AX 93\/2-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Order"],"published-print":{"date-parts":[[2024,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A poset <jats:inline-formula><jats:alternatives><jats:tex-math>$$(P^{\\prime },\\le _{P^{\\prime }})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>P<\/mml:mi>\n                      <mml:mo>\u2032<\/mml:mo>\n                    <\/mml:msup>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msub>\n                      <mml:mo>\u2264<\/mml:mo>\n                      <mml:msup>\n                        <mml:mi>P<\/mml:mi>\n                        <mml:mo>\u2032<\/mml:mo>\n                      <\/mml:msup>\n                    <\/mml:msub>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> contains a copy of some other poset <jats:inline-formula><jats:alternatives><jats:tex-math>$$(P,\\le _{P})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>P<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msub>\n                      <mml:mo>\u2264<\/mml:mo>\n                      <mml:mi>P<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> if there is an injection <jats:inline-formula><jats:alternatives><jats:tex-math>$$f:P'\\rightarrow P$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>:<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>P<\/mml:mi>\n                      <mml:mo>\u2032<\/mml:mo>\n                    <\/mml:msup>\n                    <mml:mo>\u2192<\/mml:mo>\n                    <mml:mi>P<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> where for every <jats:inline-formula><jats:alternatives><jats:tex-math>$$X,Y\\in P$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>Y<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mi>P<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$X\\le _{P} Y$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:msub>\n                      <mml:mo>\u2264<\/mml:mo>\n                      <mml:mi>P<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mi>Y<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> if and only if <jats:inline-formula><jats:alternatives><jats:tex-math>$$f(X)\\le _{P'} f(Y)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>X<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:msub>\n                      <mml:mo>\u2264<\/mml:mo>\n                      <mml:msup>\n                        <mml:mi>P<\/mml:mi>\n                        <mml:mo>\u2032<\/mml:mo>\n                      <\/mml:msup>\n                    <\/mml:msub>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>Y<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. For any posets <jats:italic>P<\/jats:italic> and <jats:italic>Q<\/jats:italic>, the poset Ramsey number <jats:italic>R<\/jats:italic>(<jats:italic>P<\/jats:italic>,\u00a0<jats:italic>Q<\/jats:italic>) is the smallest integer <jats:italic>N<\/jats:italic> such that any blue\/red coloring of a Boolean lattice of dimension <jats:italic>N<\/jats:italic> contains either a copy of <jats:italic>P<\/jats:italic> with all elements blue or a copy of <jats:italic>Q<\/jats:italic> with all elements red. A complete <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-partite poset <jats:inline-formula><jats:alternatives><jats:tex-math>$$K_{t_{1},\\dots ,t_{\\ell }}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>K<\/mml:mi>\n                    <mml:mrow>\n                      <mml:msub>\n                        <mml:mi>t<\/mml:mi>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mo>\u22ef<\/mml:mo>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>t<\/mml:mi>\n                        <mml:mi>\u2113<\/mml:mi>\n                      <\/mml:msub>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is a poset on <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\sum _{i=1}^{\\ell } t_{i}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msubsup>\n                      <mml:mo>\u2211<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mi>i<\/mml:mi>\n                        <mml:mo>=<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                      <mml:mi>\u2113<\/mml:mi>\n                    <\/mml:msubsup>\n                    <mml:msub>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mi>i<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> elements, which are partitioned into <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> pairwise disjoint sets <jats:inline-formula><jats:alternatives><jats:tex-math>$$A^{i}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>A<\/mml:mi>\n                    <mml:mi>i<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with <jats:inline-formula><jats:alternatives><jats:tex-math>$$|A^{i}|=t_{i}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:msup>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mi>i<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                      <mml:mo>=<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:msub>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mi>i<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$1\\le i\\le \\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mi>i<\/mml:mi>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mi>\u2113<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, such that for any two <jats:inline-formula><jats:alternatives><jats:tex-math>$$X\\in A^{i}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mi>i<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$Y\\in A^{j}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>Y<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mi>j<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$X&lt;Y$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mo>&lt;<\/mml:mo>\n                    <mml:mi>Y<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> if and only if <jats:inline-formula><jats:alternatives><jats:tex-math>$$i&lt;j$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>i<\/mml:mi>\n                    <mml:mo>&lt;<\/mml:mo>\n                    <mml:mi>j<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In this paper we show that <jats:inline-formula><jats:alternatives><jats:tex-math>$$R(K_{t_{1},\\dots ,t_{\\ell }},Q_{n})\\le n+\\frac{(2+o_{n}(1))\\ell n}{\\log n}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>R<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>K<\/mml:mi>\n                        <mml:mrow>\n                          <mml:msub>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:mo>,<\/mml:mo>\n                          <mml:mo>\u22ef<\/mml:mo>\n                          <mml:mo>,<\/mml:mo>\n                          <mml:msub>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mi>\u2113<\/mml:mi>\n                          <\/mml:msub>\n                        <\/mml:mrow>\n                      <\/mml:msub>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>Q<\/mml:mi>\n                        <mml:mi>n<\/mml:mi>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mfrac>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mn>2<\/mml:mn>\n                        <mml:mo>+<\/mml:mo>\n                        <mml:msub>\n                          <mml:mi>o<\/mml:mi>\n                          <mml:mi>n<\/mml:mi>\n                        <\/mml:msub>\n                        <mml:mrow>\n                          <mml:mo>(<\/mml:mo>\n                          <mml:mn>1<\/mml:mn>\n                          <mml:mo>)<\/mml:mo>\n                        <\/mml:mrow>\n                        <mml:mo>)<\/mml:mo>\n                        <mml:mi>\u2113<\/mml:mi>\n                        <mml:mi>n<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mo>log<\/mml:mo>\n                        <mml:mi>n<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:mfrac>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s11083-023-09636-8","type":"journal-article","created":{"date-parts":[[2023,5,31]],"date-time":"2023-05-31T16:03:56Z","timestamp":1685549036000},"page":"391-399","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Poset Ramsey Number $$R(P,Q_{n})$$. I. Complete Multipartite Posets"],"prefix":"10.1007","volume":"41","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7681-8859","authenticated-orcid":false,"given":"Christian","family":"Winter","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,5,31]]},"reference":[{"issue":"2","key":"9636_CR1","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1007\/s11083-016-9399-7","volume":"34","author":"M Axenovich","year":"2017","unstructured":"Axenovich, M., Walzer, S.: Boolean lattices: Ramsey properties and embeddings. Order 34(2), 287\u2013298 (2017)","journal-title":"Order"},{"key":"9636_CR2","doi-asserted-by":"crossref","unstructured":"Axenovich, M., Winter, C.: Poset Ramsey numbers: large Boolean lattice versus a fixed poset. Comb. Probab. Comput. (2023)","DOI":"10.1017\/S0963548323000032"},{"key":"9636_CR3","doi-asserted-by":"crossref","unstructured":"Bohman, T., Peng, F. A Construction for Cube Ramsey. Order, (2022)","DOI":"10.1007\/s11083-022-09613-7"},{"issue":"3","key":"9636_CR4","doi-asserted-by":"publisher","first-page":"557","DOI":"10.1007\/s11083-017-9449-9","volume":"35","author":"C Cox","year":"2018","unstructured":"Cox, C., Stolee, D.: Ramsey numbers for partially-ordered sets. Order 35(3), 557\u2013579 (2018)","journal-title":"Order"},{"issue":"4","key":"9636_CR5","doi-asserted-by":"publisher","first-page":"1097","DOI":"10.1002\/rsa.20952","volume":"57","author":"V Falgas-Ravry","year":"2020","unstructured":"Falgas-Ravry, V., Markstr\u00f6m, K., Treglown, A., Zhao, Y.: Existence thresholds and Ramsey properties of random posets. Random Struct. Algoritm. 57(4), 1097\u20131133 (2020)","journal-title":"Random Struct. Algoritm."},{"key":"9636_CR6","doi-asserted-by":"publisher","DOI":"10.1112\/blms.12767","volume-title":"Ramsey numbers of Boolean lattices","author":"D Gr\u00f3sz","year":"2023","unstructured":"Gr\u00f3sz, D., Methuku, A., Tompkins, C.: Ramsey numbers of Boolean lattices. Bull. Lond. Math, Soc (2023)"},{"issue":"2","key":"9636_CR7","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1007\/s11083-021-09557-4","volume":"39","author":"L Lu","year":"2022","unstructured":"Lu, L., Thompson, C.: Poset Ramsey numbers for boolean lattices. Order 39(2), 171\u2013185 (2022)","journal-title":"Order"},{"key":"9636_CR8","doi-asserted-by":"crossref","unstructured":"F. P. Ramsey. On a Problem of Formal Logic. Proc. Lond. Math. Soc. s2-30(1), 264\u2013286, (1930)","DOI":"10.1112\/plms\/s2-30.1.264"},{"key":"9636_CR9","unstructured":"Walzer, S. Ramsey Variant of the 2-Dimension of Posets. In: Master Thesis, Karlsruhe Institute of Technology (2015)"},{"key":"9636_CR10","unstructured":"Winter, C. Poset Ramsey Number R(P,Qn). II. Antichains. Submitted, preprint available at arXiv:2205.02275v1 (2022)"}],"container-title":["Order"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11083-023-09636-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11083-023-09636-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11083-023-09636-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,13]],"date-time":"2024-09-13T09:11:00Z","timestamp":1726218660000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11083-023-09636-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,31]]},"references-count":10,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,8]]}},"alternative-id":["9636"],"URL":"https:\/\/doi.org\/10.1007\/s11083-023-09636-8","relation":{},"ISSN":["0167-8094","1572-9273"],"issn-type":[{"type":"print","value":"0167-8094"},{"type":"electronic","value":"1572-9273"}],"subject":[],"published":{"date-parts":[[2023,5,31]]},"assertion":[{"value":"26 April 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 April 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 May 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}