{"department":[{"_id":"MaKw"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2202.05088"}],"abstract":[{"lang":"eng","text":"We prove several results about substructures in Latin squares. First, we explain how to adapt our recent work on high-girth Steiner triple systems to the setting of Latin squares, resolving a conjecture of Linial that there exist Latin squares with arbitrarily high girth. As a consequence, we see that the number of order- n  Latin squares with no intercalate (i.e., no  2×2 Latin subsquare) is at least  (e−9/4n−o(n))n2. Equivalently,  P[N=0]≥e−n2/4−o(n2)=e−(1+o(1))EN\r\n , where  N is the number of intercalates in a uniformly random order- n Latin square. \r\nIn fact, extending recent work of Kwan, Sah, and Sawhney, we resolve the general large-deviation problem for intercalates in random Latin squares, up to constant factors in the exponent: for any constant  0<δ≤1 we have  P[N≤(1−δ)EN]=exp(−Θ(n2)) and for any constant  δ>0 we have  P[N≥(1+δ)EN]=exp(−Θ(n4/3logn)). \r\nFinally, as an application of some new general tools for studying substructures in random Latin squares, we show that in almost all order- n Latin squares, the number of cuboctahedra (i.e., the number of pairs of possibly degenerate  2×2 submatrices with the same arrangement of symbols) is of order  n4, which is the minimum possible. As observed by Gowers and Long, this number can be interpreted as measuring ``how associative'' the quasigroup associated with the Latin square is."}],"publication_status":"published","date_published":"2023-09-01T00:00:00Z","issue":"2","month":"09","article_processing_charge":"Yes (in subscription journal)","date_updated":"2023-10-31T11:27:30Z","type":"journal_article","day":"01","quality_controlled":"1","title":"Substructures in Latin squares","status":"public","publisher":"Springer Nature","scopus_import":"1","oa":1,"oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2023-10-22T22:01:14Z","language":[{"iso":"eng"}],"_id":"14444","citation":{"ama":"Kwan MA, Sah A, Sawhney M, Simkin M. Substructures in Latin squares. <i>Israel Journal of Mathematics</i>. 2023;256(2):363-416. doi:<a href=\"https://doi.org/10.1007/s11856-023-2513-9\">10.1007/s11856-023-2513-9</a>","ista":"Kwan MA, Sah A, Sawhney M, Simkin M. 2023. Substructures in Latin squares. Israel Journal of Mathematics. 256(2), 363–416.","apa":"Kwan, M. A., Sah, A., Sawhney, M., &#38; Simkin, M. (2023). Substructures in Latin squares. <i>Israel Journal of Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s11856-023-2513-9\">https://doi.org/10.1007/s11856-023-2513-9</a>","chicago":"Kwan, Matthew Alan, Ashwin Sah, Mehtaab Sawhney, and Michael Simkin. “Substructures in Latin Squares.” <i>Israel Journal of Mathematics</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s11856-023-2513-9\">https://doi.org/10.1007/s11856-023-2513-9</a>.","short":"M.A. Kwan, A. Sah, M. Sawhney, M. Simkin, Israel Journal of Mathematics 256 (2023) 363–416.","mla":"Kwan, Matthew Alan, et al. “Substructures in Latin Squares.” <i>Israel Journal of Mathematics</i>, vol. 256, no. 2, Springer Nature, 2023, pp. 363–416, doi:<a href=\"https://doi.org/10.1007/s11856-023-2513-9\">10.1007/s11856-023-2513-9</a>.","ieee":"M. A. Kwan, A. Sah, M. Sawhney, and M. Simkin, “Substructures in Latin squares,” <i>Israel Journal of Mathematics</i>, vol. 256, no. 2. Springer Nature, pp. 363–416, 2023."},"article_type":"original","external_id":{"arxiv":["2202.05088"]},"publication_identifier":{"issn":["0021-2172"],"eissn":["1565-8511"]},"doi":"10.1007/s11856-023-2513-9","page":"363-416","publication":"Israel Journal of Mathematics","author":[{"orcid":"0000-0002-4003-7567","last_name":"Kwan","first_name":"Matthew Alan","id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3","full_name":"Kwan, Matthew Alan"},{"first_name":"Ashwin","last_name":"Sah","full_name":"Sah, Ashwin"},{"last_name":"Sawhney","first_name":"Mehtaab","full_name":"Sawhney, Mehtaab"},{"last_name":"Simkin","first_name":"Michael","full_name":"Simkin, Michael"}],"acknowledgement":"Sah and Sawhney were supported by NSF Graduate Research Fellowship Program DGE-1745302. Sah was supported by the PD Soros Fellowship. Simkin was supported by the Center of Mathematical Sciences and Applications at Harvard University.","year":"2023","intvolume":"       256","volume":256}