Turán $H$-Densities for 3-Graphs

Falgas–Ravry, Victor; Vaughan, Emil R.

doi:10.37236/2733

Cited by 24 publications

(39 citation statements)

References 30 publications

(58 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Flag algebras can be used as a general tool to attack problems from extremal combinatorics. Flag algebras were used for a wide range of problems, for example the Caccetta-Häggkvist conjecture [15,21], Turán-type problems in graphs [7,11,13,19,23,26,27], 3-graphs [9,10] and hypercubes [1,3], extremal problems in a colored environment [2,4,6], and also to problems in geometry [17] or extremal theory of permutations [5]. For more details on these applications, see a recent survey of Razborov [24].…”

Section: Methods and Flag Algebrasmentioning

confidence: 99%

“…For our problem, the conjectured extremal graph has an iterated structure, for which it is rare to obtain the precise density from plain flag algebra computations alone. One such rare example is the problem to determine the inducibility of small out-stars in oriented graphs [9] (note that the problem of inducibility of all out-stars was recently solved by Huang [16] using different techniques). Hladký, Krá , l and Norin announced that they found the inducibility of the oriented path of length 2, which also has an iterated extremal construction, via a flag algebra method.…”

Section: Methods and Flag Algebrasmentioning

confidence: 99%

See 1 more Smart Citation

Maximum density of induced 5-cycle is achieved by an iterated blow-up of 5-cycle

Balogh

Lidický

et al. 2016

European Journal of Combinatorics

110

View full text Add to dashboard Cite

show abstract

Section: Methods and Flag Algebrasmentioning

confidence: 99%

Section: Methods and Flag Algebrasmentioning

confidence: 99%

Maximum density of induced 5-cycle is achieved by an iterated blow-up of 5-cycle

Balogh

Lidický

et al. 2016

European Journal of Combinatorics

110

View full text Add to dashboard Cite

show abstract

“…Flag algebras have been very successful in tackling various problems. To mention some of them: Caccetta-Häggkvist conjecture [21,25,36], various Turán-type problems in graphs [10,20,22,24,29,31,32,34,37,39], hypergraphs [3,15,16,19,30] and hypercubes [2,5], extremal problems in a colored environment [4,9,23,26] and also to problems in geometry [27] or extremal theory of permutations [6]. For more details on these applications, see a survey of Razborov [35].…”

Section: Flag Algebrasmentioning

confidence: 99%

Minimum number of edges that occur in odd cycles

Grzesik

Volec

2019

Journal of Combinatorial Theory, Series B

View full text Add to dashboard Cite

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.

show abstract

“…In the case of 3-uniform hypergraphs, flag algebras have already been successfully applied to various problems. These include progress on the celebrated Turán conjecture about the minimum edge-density that forces containing K 4 as a subhypergraph [22,20,23,24], several Turán-type problems for 3-uniform hypergraphs [3,15,16], and the so-called jumps in hypergraphs [2].…”

Section: Flag Algebrasmentioning

confidence: 99%

A problem of Erdős and Sós on 3-graphs

Glebov¹,

Kráľ²,

Volec³

2013

The Seventh European Conference on Combinatorics, Graph Theory and Applications

View full text Add to dashboard Cite

Abstract. We show that for every ε > 0 there exist δ > 0 and n0 ∈ N such that every 3-uniform hypergraph on n ≥ n0 vertices with the property that every k-vertex subset, where k ≥ δn, induces at least 1 4 + ε k 3 edges, contains K − 4 as a subgraph, where K − 4 is the 3-uniform hypergraph on 4 vertices with 3 edges. This question was originally raised by Erdős and Sós. The constant 1/4 is the best possible.

show abstract

Turán $H$-Densities for 3-Graphs

Abstract: Given an r-graph H on h vertices, and a family F of forbidden subgraphs, we define ex H (n, F ) to be the maximum number of induced copies of H in an F -free r-graph on n vertices. Then the Turán H-density of F is the limit π H (F ) = lim n→∞

Cited by 24 publications

References 30 publications

Maximum density of induced 5-cycle is achieved by an iterated blow-up of 5-cycle

Maximum density of induced 5-cycle is achieved by an iterated blow-up of 5-cycle

Minimum number of edges that occur in odd cycles

A problem of Erdős and Sós on 3-graphs

Contact Info

Product

Resources

About