{"article_processing_charge":"No","month":"12","quality_controlled":"1","year":"2025","language":[{"iso":"eng"}],"volume":10,"publication_status":"published","publisher":"Alliance of Diamond Open Access Journals","type":"journal_article","intvolume":"        10","scopus_import":"1","doi":"10.19086/aic.2025.10","citation":{"mla":"Girão, António, et al. “Blowups of Triangle-Free Graphs.” <i>Advances in Combinatorics</i>, vol. 10, Alliance of Diamond Open Access Journals, 2025, doi:<a href=\"https://doi.org/10.19086/aic.2025.10\">10.19086/aic.2025.10</a>.","apa":"Girão, A., Hunter, Z., &#38; Wigderson, Y. (2025). Blowups of triangle-free graphs. <i>Advances in Combinatorics</i>. Alliance of Diamond Open Access Journals. <a href=\"https://doi.org/10.19086/aic.2025.10\">https://doi.org/10.19086/aic.2025.10</a>","chicago":"Girão, António, Zach Hunter, and Yuval Wigderson. “Blowups of Triangle-Free Graphs.” <i>Advances in Combinatorics</i>. Alliance of Diamond Open Access Journals, 2025. <a href=\"https://doi.org/10.19086/aic.2025.10\">https://doi.org/10.19086/aic.2025.10</a>.","ista":"Girão A, Hunter Z, Wigderson Y. 2025. Blowups of triangle-free graphs. Advances in Combinatorics. 10.","ieee":"A. Girão, Z. Hunter, and Y. Wigderson, “Blowups of triangle-free graphs,” <i>Advances in Combinatorics</i>, vol. 10. Alliance of Diamond Open Access Journals, 2025.","ama":"Girão A, Hunter Z, Wigderson Y. Blowups of triangle-free graphs. <i>Advances in Combinatorics</i>. 2025;10. doi:<a href=\"https://doi.org/10.19086/aic.2025.10\">10.19086/aic.2025.10</a>","short":"A. Girão, Z. Hunter, Y. Wigderson, Advances in Combinatorics 10 (2025)."},"abstract":[{"lang":"eng","text":"A highly influential result of Nikiforov states that if an n-vertex graph G contains\r\nat least γnh copies of a fixed h-vertex graph H, then G contains a blowup of H of order\r\nΩγ,H(logn). While the dependence on n is optimal, the correct dependence on γ is unknown;\r\nall known proofs yield bounds that are polynomial in γ, but the best known upper bound,\r\ncoming from random graphs, is only logarithmic in γ. It is a major open problem to narrow\r\nthis gap.\r\nWe prove that if H is triangle-free, then the logarithmic behavior of the upper bound\r\nis the truth. That is, under the assumptions above, G contains a blowup of H of order\r\nΩH(logn/log(1/γ)). This is the first non-trivial instance where the optimal dependence in\r\nNikiforov’s theorem is known.\r\nAs a consequence, we also prove an upper bound on multicolor Ramsey numbers of\r\nblowups of triangle-free graphs, proving that the dependence on the number of colors is\r\npolynomial once the blowup is sufficiently large. This shows that, from the perspective\r\nof multicolor Ramsey numbers, blowups of fixed triangle-free graphs behave like bipartite\r\ngraphs."}],"title":"Blowups of triangle-free graphs","author":[{"first_name":"António","last_name":"Girão","full_name":"Girão, António"},{"last_name":"Hunter","first_name":"Zach","full_name":"Hunter, Zach"},{"id":"2d0023a0-1567-11f0-833d-d5c1e476d4b5","full_name":"Wigderson, Yuval","first_name":"Yuval","last_name":"Wigderson"}],"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2408.12913","open_access":"1"}],"article_type":"original","OA_place":"repository","day":"19","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","date_updated":"2026-07-14T08:50:55Z","date_created":"2026-06-29T10:56:44Z","oa_version":"Preprint","_id":"22172","extern":"1","publication":"Advances in Combinatorics","date_published":"2025-12-19T00:00:00Z","arxiv":1,"OA_type":"green","oa":1,"external_id":{"arxiv":["2408.12913"]}}