Minimum Number of Monotone Subsequences of Length 4 in Permutations

Balogh, József; Hu, Ping; Lidický, Bernard; Pikhurko, Oleg; Udvari, Balázs; Volec, Jan

doi:10.1017/s0963548314000820

Cited by 31 publications

(56 citation statements)

References 33 publications

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“…Using the flag algebra semidefinite method, we were able to obtain the bound

\begin{matrix} true \\ right \forall φ \in {Hom}^{+} ({scriptA}^{0}, R), φ (W_{4}) ⩽ 0.157516, \end{matrix}

subject to floating point rounding errors. This suggests that the conjecture is true and that there may be a straightforward (but numerically intensive) proof using the semidefinite method and rounding techniques (see for some examples).…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

“…Using the flag algebra semidefinite method, we were able to obtain the bound ∀ ∈ Hom + ( 0 , ℝ), ( 4 ) ⩽ 0.157516, subject to floating point rounding errors. This suggests that the conjecture is true and that there may be a straightforward (but numerically intensive) proof using the semidefinite method and rounding techniques (see [1,10,11,14,22] for some examples). The intuition of the recursive construction of , is that at every step we have one part −1 ⧵ that maximizes the density of ⃗ 3 (hence is almost balanced) and another part whose vertices all beat the first part.…”

Section: F I G U R Ementioning

confidence: 99%

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Quasi‐carousel tournaments

Coregliano

2017

Journal of Graph Theory

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A tournament is called locally transitive if the outneighborhood and the inneighborhood of every vertex are transitive. Equivalently, a tournament is locally transitive if it avoids the tournaments W4 and L4, which are the only tournaments up to isomorphism on four vertices containing a unique 3‐cycle. On the other hand, a sequence of tournaments false(Tnfalse)n∈double-struckN with Vfalse(Tnfalse)=n is called almost balanced if all but o(n) vertices of Tn have outdegree (1/2+o(1))n. In the same spirit of quasi‐random properties, we present several characterizations of tournament sequences that are both almost balanced and asymptotically locally transitive in the sense that the density of W4 and L4 in Tn goes to zero as n goes to infinity.

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“…Using the flag algebra semidefinite method, we were able to obtain the bound

\begin{matrix} true \\ right \forall φ \in {Hom}^{+} ({scriptA}^{0}, R), φ (W_{4}) ⩽ 0.157516, \end{matrix}

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

Section: F I G U R Ementioning

confidence: 99%

Quasi‐carousel tournaments

Coregliano

2017

Journal of Graph Theory

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“…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

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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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“…Another example is an extension of the famous theorem of Erdős and Szekeres [18], which states that any sequence of more than k 2 numbers contains a monotone (that is, monotonically increasing or monotonically decreasing) subsequence of length k + 1. Here, one may ask what the minimum number of monotone subsequences of length k + 1 contained in a sequence of n numbers is; see [2,27,34]. In this paper, we consider a similar Erdős-Rademacher-type generalisation of a classical problem in additive combinatorics.…”

Section: Introductionmentioning

confidence: 99%

The number of additive triples in subsets of abelian groups

Samotij¹,

Sudakov²

2016

Math. Proc. Camb. Phil. Soc.

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A set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elements $x,y,z$ with $x+y=z$. The study of how large the largest sum-free subset of a given abelian group is had started more than thirty years before it was finally resolved by Green and Ruzsa a decade ago. We address the following more general question. Suppose that a set $A$ of elements of an abelian group $G$ has cardinality $a$. How many Schur triples must $A$ contain? Moreover, which sets of $a$ elements of $G$ have the smallest number of Schur triples? In this paper, we answer these questions for various groups $G$ and ranges of $a$.Comment: 20 pages; corrected the erroneous equality in (1) in the statement of Theorem 1.

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Minimum Number of Monotone Subsequences of Length 4 in Permutations

Cited by 31 publications

References 33 publications

Quasi‐carousel tournaments

Quasi‐carousel tournaments

Minimum number of edges that occur in odd cycles

The number of additive triples in subsets of abelian groups

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