{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,29]],"date-time":"2025-12-29T13:23:48Z","timestamp":1767014628171,"version":"3.48.0"},"reference-count":19,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T00:00:00Z","timestamp":1764547200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,12,3]],"date-time":"2025-12-03T00:00:00Z","timestamp":1764720000000},"content-version":"vor","delay-in-days":2,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    The following question was asked by Prendiville: given an\n                    <jats:italic>r<\/jats:italic>\n                    -colouring of the interval\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\{2, \\dotsc , N\\}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mo>{<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\u22ef<\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo>}<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , what is the minimum number of monochromatic solutions of the equation\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$xy = z$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mi>y<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>z<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ? For\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$r=2$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , we show that there are always asymptotically at least\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$(1\/2\\sqrt{2}) N^{1\/2} \\log N$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>\/<\/mml:mo>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:msqrt>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:msqrt>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msup>\n                              <mml:mi>N<\/mml:mi>\n                              <mml:mrow>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>\/<\/mml:mo>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>log<\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    monochromatic solutions, and that the leading constant is sharp. For\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$r=3$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$r=4$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>4<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    we obtain tight results up to a multiplicative logarithmic factor. We also provide bounds for more colours and other multiplicative equations.\n                  <\/jats:p>","DOI":"10.1007\/s00493-025-00183-x","type":"journal-article","created":{"date-parts":[[2025,12,3]],"date-time":"2025-12-03T08:58:50Z","timestamp":1764752330000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the number of monochromatic solutions to multiplicative equations"],"prefix":"10.1007","volume":"45","author":[{"given":"Lucas","family":"Arag\u00e3o","sequence":"first","affiliation":[]},{"given":"Jonathan","family":"Chapman","sequence":"additional","affiliation":[]},{"given":"Miquel","family":"Ortega","sequence":"additional","affiliation":[]},{"given":"Victor","family":"Souza","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,12,3]]},"reference":[{"key":"183_CR1","doi-asserted-by":"crossref","unstructured":"Abbott, H., Hanson, D.: A problem of Schur and its generalizations. Acta Arith 20(2), 175\u2013187 (1972)","DOI":"10.4064\/aa-20-2-175-187"},{"key":"183_CR2","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1007\/978-3-0348-5438-2_19","volume-title":"Studies in Pure Mathematics: To the Memory of Paul Tur\u00e1n","author":"P Erd\u0151s","year":"1983","unstructured":"Erd\u0151s, P., Szemer\u00e9di, E.: On sums and products of integers. In: Erd\u0151s, P., Alp\u00e1r, L., Hal\u00e1sz, G., S\u00e1rk\u00f6zy, A. (eds.) Studies in Pure Mathematics: To the Memory of Paul Tur\u00e1n, pp. 213\u2013218. Birkh\u00e4user Basel, Basel (1983)"},{"key":"183_CR3","doi-asserted-by":"crossref","unstructured":"Exoo, G.: A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers. Electronic Journal of Combinatorics, R8\u2013R8 (1994)","DOI":"10.37236\/1188"},{"issue":"2","key":"183_CR4","doi-asserted-by":"publisher","first-page":"367","DOI":"10.4007\/annals.2008.168.367","volume":"168","author":"K Ford","year":"2008","unstructured":"Ford, K.: The distribution of integers with a divisor in a given interval. Ann. Math. 168(2), 367\u2013433 (2008)","journal-title":"Ann. Math."},{"issue":"2","key":"183_CR5","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1016\/0097-3165(88)90020-9","volume":"47","author":"P Frankl","year":"1988","unstructured":"Frankl, P., Graham, R.L., R\u00f6dl, V.: Quantitative theorems for regular systems of equations. Journal of Combinatorial Theory. Series A 47(2), 246\u2013261 (1988)","journal-title":"Journal of Combinatorial Theory. Series A"},{"issue":"8","key":"183_CR6","first-page":"49","volume":"4","author":"L Funar","year":"1990","unstructured":"Funar, L.: Generalized sum-free sets of integers. Nieuw Archief voor Wiskunde 4(8), 49\u201354 (1990)","journal-title":"Nieuw Archief voor Wiskunde"},{"key":"183_CR7","doi-asserted-by":"crossref","unstructured":"Graham, R.L., R\u00f6dl, V., Ruci\u0144ski, A.: On Schur properties of random subsets of integers. Journal of Number Theory 61(2), 388\u2013408 (1996)","DOI":"10.1006\/jnth.1996.0155"},{"key":"183_CR8","unstructured":"Graham, R. L., Rothschild, B. L., Spencer, J. H.: Ramsey Theory. Second edition. Wiley Series in Discrete Mathematics and Optimization. Wiley, (2013)"},{"key":"183_CR9","doi-asserted-by":"crossref","unstructured":"Heule, M.: Schur Number Five. Proceedings of the AAAI Conference on Artificial Intelligence 32(1) (2018)","DOI":"10.1609\/aaai.v32i1.12209"},{"issue":"2","key":"183_CR10","doi-asserted-by":"publisher","first-page":"1082","DOI":"10.1137\/24M1632875","volume":"39","author":"L Mattos","year":"2025","unstructured":"Mattos, L., Mergoni Cecchelli, D., Parczyk, O.: On product Schur triples in the integers. SIAM Journal on Discrete Mathematics 39(2), 1082\u20131095 (2025)","journal-title":"SIAM Journal on Discrete Mathematics"},{"key":"183_CR11","doi-asserted-by":"crossref","unstructured":"Montgomery, H. L., Vaughan, R. C.: Multiplicative Number Theory I: Classical Theory. Vol. 97. Cambridge Studies in Advanced Mathematics. Cambridge University Press, (2007)","DOI":"10.1017\/CBO9780511618314"},{"key":"183_CR12","unstructured":"Prendiville, S.: Counting monochromatic solutions to diagonal Diophantine equations. Discrete Analysis 14 (2021)"},{"issue":"1","key":"183_CR13","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1007\/BF01188632","volume":"36","author":"R Rado","year":"1933","unstructured":"Rado, R.: Studien zur kombinatorik. Math. Z. 36(1), 424\u2013470 (1933)","journal-title":"Math. Z."},{"key":"183_CR14","doi-asserted-by":"crossref","unstructured":"Robertson, A., Zeilberger, D.: A 2-Coloring of $$[1,N]$$ can have $$(1\/22)N^2+O(N)$$ Monochromatic Schur Triples, but not less! Electronic Journal of Combinatorics, R19 (1998)","DOI":"10.37236\/1357"},{"issue":"2","key":"183_CR15","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1017\/S0305004121000633","volume":"173","author":"M Rudnev","year":"2022","unstructured":"Rudnev, M., Stevens, S.: An update on the sum-product problem. Math. Proc. Cambridge Philos. Soc. 173(2), 411\u2013430 (2022)","journal-title":"Math. Proc. Cambridge Philos. Soc."},{"key":"183_CR16","doi-asserted-by":"crossref","unstructured":"Schoen, T.: The number of monochromatic Schur triples. Eur. J. Comb. 20(8), 855\u2013866 (1999)","DOI":"10.1006\/eujc.1999.0297"},{"key":"183_CR17","first-page":"114","volume":"25","author":"I Schur","year":"1916","unstructured":"Schur, I.: \u00dcber die kongruenz $$x^m+ y^m \\equiv z^m ~(\\text{ mod }\\; p)$$. Jahresber. Deutsch. Math.-Verein. 25, 114\u2013117 (1916)","journal-title":"Jahresber. Deutsch. Math.-Verein."},{"issue":"04","key":"183_CR18","doi-asserted-by":"publisher","first-page":"491","DOI":"10.1112\/S0024609305004261","volume":"37","author":"J Solymosi","year":"2005","unstructured":"Solymosi, J.: On the number of sums and products. Bull. Lond. Math. Soc. 37(04), 491\u2013494 (2005)","journal-title":"Bull. Lond. Math. Soc."},{"key":"183_CR19","first-page":"212","volume":"19","author":"BL van der Waerden","year":"1927","unstructured":"van der Waerden, B.L.: Beweis einer baudetschen vermutung. Nieuw Archief voor Wiskunde. Vijfde Serie 19, 212\u2013216 (1927)","journal-title":"Nieuw Archief voor Wiskunde. Vijfde Serie"}],"container-title":["Combinatorica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-025-00183-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00493-025-00183-x","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-025-00183-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,29]],"date-time":"2025-12-29T13:17:57Z","timestamp":1767014277000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00493-025-00183-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12]]},"references-count":19,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["183"],"URL":"https:\/\/doi.org\/10.1007\/s00493-025-00183-x","relation":{},"ISSN":["0209-9683","1439-6912"],"issn-type":[{"type":"print","value":"0209-9683"},{"type":"electronic","value":"1439-6912"}],"subject":[],"published":{"date-parts":[[2025,12]]},"assertion":[{"value":"10 August 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 July 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 August 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 December 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"64"}}