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The codegree threshold of K_4^-PrimeFaces.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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2017 (English)In: The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17) / [ed] Drmota Michael; Kang Mihyun; Krattenthaler Christian; Nešetřil Jaroslav, Elsevier, 2017, Vol. 61, p. 407-413Conference paper, Published paper (Refereed)
##### Abstract [en]

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

Elsevier, 2017. Vol. 61, p. 407-413
##### Series

Electronic Notes in Discrete Mathematics, ISSN 1571-0653
##### Keywords [en]

extremal combinatorics, hypergraphs, codegree treshold, flag algebras
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:umu:diva-144158DOI: 10.1016/j.endm.2017.06.067OAI: oai:DiVA.org:umu-144158DiVA, id: diva2:1176928
##### Conference

EUROCOMB 2017, The European Conference on Combinatorics, Graph Theory and Applications, Vienna, Italy, August 28 - September 1, 2017
#####

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##### Funder

Swedish Research CouncilAvailable from: 2018-01-23 Created: 2018-01-23 Last updated: 2018-06-14Bibliographically approved

The codegree threshold ex2(n, F) of a non-empty 3-graph F is the minimum d = d(n) such that every 3-graph on n vertices in which every pair of vertices is contained in at least d+ 1 edges contains a copy of F as a subgraph. We study ex2(n, F) when F = K − 4 , the 3-graph on 4 vertices with 3 edges. Using flag algebra techniques, we prove that

ex2(n, K− 4 ) = n 4 + o(n).

This settles in the affirmative a conjecture of Nagle [20]. In addition, we obtain a stability result: for every near-extremal configurations G, there is a quasirandom tournament T on the same vertex set such that G is close in the edit distance to the 3-graph C(T) whose edges are the cyclically oriented triangles from T. For infinitely many values of n, we are further able to determine ex2(n, K− 4 ) exactly and to show that tournament-based constructions C(T) are extremal for those values of n.

doi
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