{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article11536","name":"On Minrank and Forbidden Subgraphs","abstract":"The minrank over a field F of a graph G on the vertex set {1,2,...,n} is the minimum possible rank of a matrix M in F^{n x n} such that M_{i,i} != 0 for every i, and M_{i,j}=0 for every distinct non-adjacent vertices i and j in G. For an integer n, a graph H, and a field F, let g(n,H,F) denote the maximum possible minrank over F of an n-vertex graph whose complement contains no copy of H. In this paper we study this quantity for various graphs H and fields F. For finite fields, we prove by a probabilistic argument a general lower bound on g(n,H,F), which yields a nearly tight bound of Omega(sqrt{n}\/log n) for the triangle H=K_3. For the real field, we prove by an explicit construction that for every non-bipartite graph H, g(n,H,R) >= n^delta for some delta = delta(H)>0. As a by-product of this construction, we disprove a conjecture of Codenotti, Pudl\u00e1k, and Resta. The results are motivated by questions in information theory, circuit complexity, and geometry.","keywords":["Minrank","Forbidden subgraphs","Shannon capacity","Circuit Complexity"],"author":{"@type":"Person","name":"Haviv, Ishay","givenName":"Ishay","familyName":"Haviv","affiliation":"School of Computer Science, The Academic College of Tel Aviv-Yaffo, Tel Aviv 61083, Israel"},"position":42,"pageStart":"42:1","pageEnd":"42:14","dateCreated":"2018-08-13","datePublished":"2018-08-13","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Haviv, Ishay","givenName":"Ishay","familyName":"Haviv","affiliation":"School of Computer Science, The Academic College of Tel Aviv-Yaffo, Tel Aviv 61083, Israel"},"copyrightYear":"2018","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.APPROX-RANDOM.2018.42","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6319","volumeNumber":116,"name":"Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX\/RANDOM 2018)","dateCreated":"2018-08-13","datePublished":"2018-08-13","editor":[{"@type":"Person","name":"Blais, Eric","givenName":"Eric","familyName":"Blais"},{"@type":"Person","name":"Jansen, Klaus","givenName":"Klaus","familyName":"Jansen"},{"@type":"Person","name":"D. P. Rolim, Jos\u00e9","givenName":"Jos\u00e9","familyName":"D. P. Rolim"},{"@type":"Person","name":"Steurer, David","givenName":"David","familyName":"Steurer"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article11536","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6319"}}}