{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:37:45Z","timestamp":1760402265553,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,17]],"date-time":"2020-04-17T00:00:00Z","timestamp":1587081600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Research Foundation of Korea(NRF)","award":["2017R1A6A3A04005963"],"award-info":[{"award-number":["2017R1A6A3A04005963"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A family    F    is an intersecting family if any two members have a nonempty intersection. Erd\u0151s, Ko, and Rado showed that      | F |  \u2264    n \u2212 1   k \u2212 1        holds for a k-uniform intersecting family    F    of subsets of     [ n ]    . The Erd\u0151s-Ko-Rado theorem for non-uniform intersecting families of subsets of     [ n ]     of size at most k can be easily proved by applying the above result to each uniform subfamily of a given family. It establishes that      | F |  \u2264    n \u2212 1   k \u2212 1    +    n \u2212 1   k \u2212 2    + \u22ef +    n \u2212 1  0       holds for non-uniform intersecting families of subsets of     [ n ]     of size at most k. In this paper, we prove that the same upper bound of the Erd\u0151s-Ko-Rado Theorem for k-uniform intersecting families of subsets of     [ n ]     holds also in the non-uniform family of subsets of     [ n ]     of size at least k and at most     n \u2212 k     with one more additional intersection condition. Our proof is based on the method of linearly independent polynomials.<\/jats:p>","DOI":"10.3390\/sym12040640","type":"journal-article","created":{"date-parts":[[2020,4,21]],"date-time":"2020-04-21T05:48:52Z","timestamp":1587448132000},"page":"640","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Erd\u0151s-Ko-Rado Type Theorem via the Polynomial Method"],"prefix":"10.3390","volume":"12","author":[{"given":"Kyung-Won","family":"Hwang","sequence":"first","affiliation":[{"name":"Department of Mathematics, Dong-A University, Busan 49315, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Younjin","family":"Kim","sequence":"additional","affiliation":[{"name":"Institute of Mathematical Sciences, Ewha Womans University, Seoul 03760, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Naeem N.","family":"Sheikh","sequence":"additional","affiliation":[{"name":"School of Sciences and Engineering, Al Akhawayn University, 53000 Ifrane, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1093\/qmath\/12.1.313","article-title":"Intersection theorem for systems of finite sets","volume":"12","author":"Ko","year":"1961","journal-title":"Q. J. Math. Oxf. Ser."},{"key":"ref_2","first-page":"365","article-title":"The Erd\u0151s-Ko-Rado theorem is true for n = ckt","volume":"Volume 1","author":"Frankl","year":"1978","journal-title":"Combinatorics, Proceedings of the Fifth Hungarian Colloquium, Keszthely"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/BF02579226","article-title":"The exact bound in the Erd\u0151s-Ko-Rado theorem","volume":"4","author":"Wilson","year":"1984","journal-title":"Combinatorica"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1137\/130947295","article-title":"Maximizing the number of nonnegative subsets","volume":"28","author":"Alon","year":"2014","journal-title":"SIAM J. Discret. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1137\/0604042","article-title":"Erd\u0151s-Ko-Rado theorem\u201322 years later","volume":"4","author":"Deza","year":"1983","journal-title":"SIAM J. Algebr. Discret. Methods"},{"key":"ref_6","first-page":"737","article-title":"On t-designs","volume":"12","author":"Wilson","year":"1975","journal-title":"Osaka J. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1007\/BF02579457","article-title":"Intersection theorems with geometric consequences","volume":"1","author":"Frankl","year":"1981","journal-title":"Combinatorica"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/BF02579189","article-title":"On functions of strength t","volume":"3","author":"Deza","year":"1983","journal-title":"Combinatorica"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/0097-3165(91)90058-O","article-title":"Multilinear polynomials and Frankl-Ray-Chaudhuri-Wilson type intersection theorems","volume":"58","author":"Alon","year":"1991","journal-title":"J. Comb. Theory Ser. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.ejc.2014.08.027","article-title":"A proof of Alon-Babai-Suzuki\u2019s Conjecture and Multilinear Polynomials","volume":"43","author":"Hwang","year":"2015","journal-title":"Eur. J. Comb."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1007\/BF02128673","article-title":"Solution of an extremal problem for sets using resultants of polynomials","volume":"10","author":"Blokhuis","year":"1990","journal-title":"Combinatorica"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.jcta.2008.04.008","article-title":"Set systems with L-intersections modulo a prime number","volume":"116","author":"Chen","year":"2009","journal-title":"J. Comb. Theory Ser. A"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"F\u00fcredi, Z., Hwang, K., and Weichsel, P. (2006). A proof and generalization of the Er\u0151s-Ko-Rado theorem using the method of linearly independent polynomials. Topics in Discrete Mathematics, Springer. Algorithms Combin. 26.","DOI":"10.1007\/3-540-33700-8_13"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1016\/j.ejc.2013.10.006","article-title":"Set systems with restricted k-wise L-intersections modulo a prime number","volume":"36","author":"Liu","year":"2014","journal-title":"Eur. J. Comb."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1023\/A:1008715718935","article-title":"On mod p Alon-Babai-Suzuki inequality","volume":"12","author":"Qian","year":"2000","journal-title":"J. Algebr. Comb."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1006\/jcta.1997.2774","article-title":"Proof of a conjecture of Frankl and F\u00fcredi","volume":"79","author":"Ramanan","year":"1997","journal-title":"J. Comb. Theory Ser. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1016\/0097-3165(94)90103-1","article-title":"On generalizations of the de Bruijn-Erd\u0151s theorem","volume":"68","author":"Snevily","year":"1994","journal-title":"J. Comb. Theory Ser. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1007\/s00493-003-0031-2","article-title":"A sharp bound for the number of sets that pairwise intersect at k positive values","volume":"23","author":"Snevily","year":"2003","journal-title":"Combinatorica"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.disc.2017.08.019","article-title":"A strengthened inequality of Alon-Babai-Suzuki\u2019s conjecture on set systems with restricted intersections modulo p","volume":"341","author":"Wang","year":"2018","journal-title":"Discret. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/640\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:31:24Z","timestamp":1760362284000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/640"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,17]]},"references-count":19,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["sym12040640"],"URL":"https:\/\/doi.org\/10.3390\/sym12040640","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,4,17]]}}}