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Applications of the semi-definite method to the Turan density problem for 3-graphsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2013 (English)In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 22, no 1, 21-54 p.Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

NEW YORK, NY, USA: Cambridge University Press, 2013. Vol. 22, no 1, 21-54 p.
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:umu:diva-63752DOI: 10.1017/S0963548312000508ISI: 000312036300003OAI: oai:DiVA.org:umu-63752DiVA: diva2:585610
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Available from: 2013-01-10 Created: 2013-01-07 Last updated: 2013-01-10Bibliographically approved

In this paper, we prove several new Turan density results for 3-graphs with independent neighbourhoods. We show: pi(K-4, C-5, F-3,F-2) = 12/49, pi(K-4, F-3,F-2) = 5/18 and pi(J(4), F-3,F-2) = pi(J(5), F-3,F-2) = 3/8, where J(t) is the 3-graph consisting of a single vertex x together with a disjoint set A of size t and all (vertical bar A vertical bar 2) 3-edges containing x. We also prove two Turan density results where we forbid certain induced subgraphs: pi(F-3,F-2, induced K-4(-)) = 3/8 and pi(K-5, 5-set spanning exactly 8 edges) = 3/4. The latter result is an analogue for K-5 of Razborov's result that pi(K-4, 4-set spanning exactly 1 edge) = 5/9. We give several new constructions, conjectures and bounds for Turan densities of 3-graphs which should be of interest to researchers in the area. Our main tool is 'Flagmatic', an implementation of Razborov's semi-definite method, which we are making publicly available. In a bid to make the power of Razborov's method more widely accessible, we have tried to make Flagmatic as user-friendly as possible, hoping to remove thereby the major hurdle that needs to be cleared before using the semi-definite method. Finally, we spend some time reflecting on the limitations of our approach, and in particular on which problems we may be unable to solve. Our discussion of the 'complexity barrier' for the semi-definite method may be of general interest.

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