The Codegree Threshold for 3-Graphs with Independent Neighborhoods

Falgas–Ravry, Victor; Marchant, Edward; Pikhurko, Oleg; Vaughan, Emil R.

doi:10.1137/130926997

Cited by 22 publications

(32 citation statements)

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“…(See [15] for details and definitions.) Marchant, Pikhurko, Vaughan and the author [5] determined the codegree threshold of F 3,2 = ( [5], {123, 124, 125, 345}), while Pikhurko,Vaughan and the author determined the codegree density of K − 4 = ( [4], {123, 124, 134}), resolving a conjecture of Nagle [16]. Nagle [16] and Czygrinow and Nagle [3] have in addition conjectured that γ(K 4 ) = 1/2, where K 4 denotes the complete 3-graph on 4 vertices.…”

Section: Historymentioning

confidence: 99%

On the Codegree Density of Complete 3-Graphs and Related Problems

Falgas–Ravry

2013

Electron. J. Combin.

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Given a family of 3-graphs F , its codegree threshold coex(n, F ) is the largest number d = d(n) such that there exists an n-vertex 3-graph in which every pair of vertices is contained in at least d 3-edges but which contains no member of F as a subgraph. The codegree density of F is the limitIn this paper we generalise a construction of Czygrinow and Nagle to bound below the codegree density of complete 3-graphs: for all integers s ≥ 4, the codegree density of the complete 3-graph on s vertices K s satisfiesWe also provide constructions based on Steiner triple systems which show that if this lower bound is sharp, then we do not have stability in general.In addition we prove bounds on the codegree density for two other infinite families of 3-graphs.

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Section: Historymentioning

confidence: 99%

On the Codegree Density of Complete 3-Graphs and Related Problems

Falgas–Ravry

2013

Electron. J. Combin.

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“…In other words, the density of red edges is at least 2 + √ 2 /8. The moreover part of the lemma follows from a standard O n −1 error estimate in the semidefinite method (for details, see, for example, [14]), and the O n −1/8 estimate on the densities of B 3 and B + 3 in G.…”

Section: Case 1 -Graphs With Many Trianglesmentioning

confidence: 99%

Minimum number of edges that occur in odd cycles

Grzesik

Volec

2019

Journal of Combinatorial Theory, Series B

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If a graph G has n ≥ 4k vertices and more than n 2 /4 edges, then it contains a copy of C 2k+1 . In 1992, Erdős, Faudree and Rousseau showed even more, that the number of edges that occur in a triangle of such a G is at least 2 n/2 + 1, and this bound is tight. They also showed that the minimum number of edges in G that occur in a copy of C 2k+1 for k ≥ 2 suddenly starts being quadratic in n, and conjectured that for any k ≥ 2, the correct lower bound should be 2n 2 /9 − O(n). Very recently, Füredi and Maleki constructed a counterexample for k = 2 and proved an asymptotically matching lower bound, namely that for any ε > 0 graphs with (1 + ε)n 2 /4 edges contain at least (2 + √ 2)n 2 /16 ∼ 0.2134n 2 edges that occur in C 5 . In this paper, we use a different approach to tackle this problem and prove the following stronger result: Every n-vertex graph with at least n 2 /4 +1 edges has at least (2+ √ 2)n 2 /16− O(n 15/8 ) edges that occur in C 5 . Next, for all k ≥ 3 and n sufficiently large, we determine the exact minimum number of edges that occur in C 2k+1 for n-vertex graphs with more than n 2 /4 edges, and show it is indeed equal to n 2 4 + 1 − n+4 6 n+1 6 = 2n 2 /9 − O(n). For both of these results, we give a structural description of all the large extremal configurations as well as obtain the corresponding stability results, which answers a conjecture of Füredi and Maleki.The main ingredient of our results is a novel approach that combines the semidefinite method from flag algebras together with ideas from finite forcibility of graph limits, which may be of independent interest. This approach allowed us to keep track of the additional extra edge needed to guarantee even the existence of a single copy of C 2k+1 , which a standard flag algebra approach would not be able to handle. Also, we establish the first application of the semidefinite method in a setting, where the set of tight examples has exponential size and arises from two very different constructions.

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“…The lower bound is from the codegree density of F 3,2 . An F 3,2 -free construction on n vertices with codegree ⌊ n 3 ⌋ − 1 is obtained by considering a tripartition of [n] into three parts with sizes as equal as possible, |V 3 | − 1 ≤ |V 1 | ≤ |V 2 | ≤ |V 3 | and adding all triples of the form V i V i V i+1 (this is not actually best possible -see [7] for a determination of the precise codegree threshold and the extremal constructions attaining it).…”

Section: F 32mentioning

confidence: 99%