{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T20:12:21Z","timestamp":1774037541715,"version":"3.50.1"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,3,27]],"date-time":"2025-03-27T00:00:00Z","timestamp":1743033600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,3,27]],"date-time":"2025-03-27T00:00:00Z","timestamp":1743033600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"EPFL Lausanne"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2025,4]]},"abstract":"Abstract<\/jats:title>\n For two graphs F<\/jats:italic>,\u00a0H<\/jats:italic> and a positive integer n<\/jats:italic>, the function \n \n $$f_{F,H}(n)$$<\/jats:tex-math>\n \n \n \n f<\/mml:mi>\n \n F<\/mml:mi>\n ,<\/mml:mo>\n H<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> denotes the largest m<\/jats:italic> such that every H<\/jats:italic>-free graph on n<\/jats:italic> vertices contains an F<\/jats:italic>-free induced subgraph on m<\/jats:italic> vertices. This function has been extensively studied in the last 60 years when F<\/jats:italic> and H<\/jats:italic> are cliques and became known as the Erd\u0151s\u2013Rogers function. Recently, Balogh, Chen and Luo, and Mubayi and Verstra\u00ebte initiated the systematic study of this function in the case where F<\/jats:italic> is a general graph. Answering, in a strong form, a question of Mubayi and Verstra\u00ebte, we prove that for every positive integer r<\/jats:italic> and every \n \n $$K_{r-1}$$<\/jats:tex-math>\n \n \n K<\/mml:mi>\n \n r<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-free graph F<\/jats:italic>, there exists some \n \n $$\\varepsilon _F>0$$<\/jats:tex-math>\n \n \n \n \u03b5<\/mml:mi>\n F<\/mml:mi>\n <\/mml:msub>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> such that \n \n $$f_{F,K_r}(n)=O(n^{1\/2-\\varepsilon _F})$$<\/jats:tex-math>\n \n \n \n f<\/mml:mi>\n \n F<\/mml:mi>\n ,<\/mml:mo>\n \n K<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n O<\/mml:mi>\n \n (<\/mml:mo>\n \n n<\/mml:mi>\n \n 1<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n -<\/mml:mo>\n \n \u03b5<\/mml:mi>\n F<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. This result is tight in two ways. Firstly, it is no longer true if F<\/jats:italic> contains \n \n $$K_{r-1}$$<\/jats:tex-math>\n \n \n K<\/mml:mi>\n \n r<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> as a subgraph. Secondly, we show that for all \n \n $$r\\ge 4$$<\/jats:tex-math>\n \n \n r<\/mml:mi>\n \u2265<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n $$\\varepsilon >0$$<\/jats:tex-math>\n \n \n \u03b5<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, there exists a \n \n $$K_{r-1}$$<\/jats:tex-math>\n \n \n K<\/mml:mi>\n \n r<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-free graph F<\/jats:italic> for which \n \n $$f_{F,K_r}(n)=\\Omega (n^{1\/2-\\varepsilon })$$<\/jats:tex-math>\n \n \n \n f<\/mml:mi>\n \n F<\/mml:mi>\n ,<\/mml:mo>\n \n K<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \u03a9<\/mml:mi>\n \n (<\/mml:mo>\n \n n<\/mml:mi>\n \n 1<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n -<\/mml:mo>\n \u03b5<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. Along the way of proving this, we show in particular that for every graph F<\/jats:italic> with minimum degree t<\/jats:italic>, we have \n \n $$f_{F,K_4}(n)=\\Omega (n^{1\/2-6\/\\sqrt{t}})$$<\/jats:tex-math>\n \n \n \n f<\/mml:mi>\n \n F<\/mml:mi>\n ,<\/mml:mo>\n \n K<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \u03a9<\/mml:mi>\n \n (<\/mml:mo>\n \n n<\/mml:mi>\n \n 1<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n -<\/mml:mo>\n 6<\/mml:mn>\n \/<\/mml:mo>\n \n t<\/mml:mi>\n <\/mml:msqrt>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. This answers (in a strong form) another question of Mubayi and Verstra\u00ebte. Finally, we prove that there exist absolute constants \n \n $$0<c<C$$<\/jats:tex-math>\n \n \n 0<\/mml:mn>\n <<\/mml:mo>\n c<\/mml:mi>\n <<\/mml:mo>\n C<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> such that for each \n \n $$r\\ge 4$$<\/jats:tex-math>\n \n \n r<\/mml:mi>\n \u2265<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, if F<\/jats:italic> is a bipartite graph with sufficiently large minimum degree, then \n \n $$\\Omega (n^{\\frac{c}{\\log r}})\\le f_{F,K_r}(n)\\le O(n^{\\frac{C}{\\log r}})$$<\/jats:tex-math>\n \n \n \u03a9<\/mml:mi>\n \n (<\/mml:mo>\n \n n<\/mml:mi>\n \n c<\/mml:mi>\n \n log<\/mml:mo>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2264<\/mml:mo>\n \n f<\/mml:mi>\n \n F<\/mml:mi>\n ,<\/mml:mo>\n \n K<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2264<\/mml:mo>\n O<\/mml:mi>\n \n (<\/mml:mo>\n \n n<\/mml:mi>\n \n C<\/mml:mi>\n \n log<\/mml:mo>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. This shows that for graphs F<\/jats:italic> with large minimum degree, the behaviour of \n \n $$f_{F,K_r}(n)$$<\/jats:tex-math>\n \n \n \n f<\/mml:mi>\n \n F<\/mml:mi>\n ,<\/mml:mo>\n \n K<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is drastically different from that of the corresponding off-diagonal Ramsey number \n \n $$f_{K_2,K_r}(n)$$<\/jats:tex-math>\n \n \n \n f<\/mml:mi>\n \n \n K<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n K<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00493-025-00147-1","type":"journal-article","created":{"date-parts":[[2025,3,30]],"date-time":"2025-03-30T06:05:51Z","timestamp":1743314751000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Induced Subgraphs of $$K_r$$-Free Graphs and the Erd\u0151s\u2013Rogers Problem"],"prefix":"10.1007","volume":"45","author":[{"given":"Lior","family":"Gishboliner","sequence":"first","affiliation":[]},{"given":"Oliver","family":"Janzer","sequence":"additional","affiliation":[]},{"given":"Benny","family":"Sudakov","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,3,27]]},"reference":[{"key":"147_CR1","doi-asserted-by":"publisher","first-page":"354","DOI":"10.1016\/0097-3165(80)90030-8","volume":"29","author":"M Ajtai","year":"1980","unstructured":"Ajtai, M., Koml\u00f3s, J., Szemer\u00e9di, E.: A note on Ramsey numbers. 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