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For a fixed graph\n G<\/jats:italic>\n , we are interested in the most number\n \n \n $$\\textrm{vex}(n,G)$$<\/jats:tex-math>\n \n \n vex<\/mml:mtext>\n (<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n G<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of vertices of\n \n \n $$Kn_n$$<\/jats:tex-math>\n \n \n K<\/mml:mi>\n \n n<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n that span a\n G<\/jats:italic>\n -free subgraph in\n \n \n $$Kn_n$$<\/jats:tex-math>\n \n \n K<\/mml:mi>\n \n n<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We show that the asymptotics of\n \n \n $$\\textrm{vex}(n,G)$$<\/jats:tex-math>\n \n \n vex<\/mml:mtext>\n (<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n G<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is\n \n \n $$(1+o(1))2^{n-1}$$<\/jats:tex-math>\n \n \n \n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n o<\/mml:mi>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \n 2<\/mml:mn>\n \n n<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for bipartite\n G<\/jats:italic>\n and\n \n \n $$(1-o(1))2^n$$<\/jats:tex-math>\n \n \n \n (<\/mml:mo>\n 1<\/mml:mn>\n -<\/mml:mo>\n o<\/mml:mi>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \n 2<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for graphs with chromatic number at least 3. We also obtain results on the order of magnitude of\n \n \n $$2^{n-1}-\\!\\textrm{vex}(n,G)$$<\/jats:tex-math>\n \n \n \n 2<\/mml:mn>\n \n n<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n -<\/mml:mo>\n \n vex<\/mml:mtext>\n \n (<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n G<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n $$2^n-\\!\\textrm{vex}(n,G)$$<\/jats:tex-math>\n \n \n \n 2<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msup>\n -<\/mml:mo>\n \n vex<\/mml:mtext>\n \n (<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n G<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in these two cases. In the case of bipartite\n G<\/jats:italic>\n , we relate this problem to instances of the forbidden subposet problem.\n <\/jats:p>","DOI":"10.1007\/s00373-026-03013-z","type":"journal-article","created":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T05:33:44Z","timestamp":1769751224000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A note on vertex Tur\u00e1n problems in the Kneser cube"],"prefix":"10.1007","volume":"42","author":[{"given":"D\u00e1niel","family":"Gerbner","sequence":"first","affiliation":[]},{"given":"Bal\u00e1zs","family":"Patk\u00f3s","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,1,30]]},"reference":[{"key":"3013_CR1","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1016\/j.jcta.2018.06.010","volume":"159","author":"M Alishahi","year":"2018","unstructured":"Alishahi, M., Taherkhani, A.: Extremal $$G$$-free induced subgraphs of Kneser graphs. 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