Variations on Twins in Permutations

Let $\pi$ be a permutation of the set $[n]=\{1,2,\dots, n\}$. Two disjoint order-isomorphic subsequences of $\pi$ are called twins. How long twins are contained in every permutation? The well known Erdős-Szekeres theorem implies that there is always a pair of twins of length $\Omega(\sqrt{n})$. On the other hand, by a simple probabilistic argument Gawron proved that for every $n\geqslant 1$ there exist permutations with all twins having length $O(n^{2/3})$. He conjectured that the latter bound is the correct size of the longest twins guaranteed in every permutation. We support this conjecture by showing that almost all permutations contain twins of length $\Omega(n^{2/3}/\log n^{1/3})$. Recently, Bukh and Rudenko have tweaked our proof and removed the log-factor. For completeness, we also present our version of their proof (see Remark 2 below on the interrelation between the two proofs). In addition, we study several variants of the problem with diverse restrictions imposed on the twins. For instance, if we restrict attention to twins avoiding a fixed permutation $\tau$, then the corresponding extremal function equals $\Theta(\sqrt{n})$, provided that $\tau$ is not monotone. In case of block twins (each twin occupies a segment) we prove that it is $(1+o(1))\frac{\log n}{\log\log n}$, while for random permutations it is twice as large. For twins that jointly occupy a segment (tight twins), we prove that for every $n$ there are permutations avoiding them on all segments of length greater than $24$.

Delta-extremal Functions in Cn

The article is devoted to properties of a weighted Green function. We study the (δ, ψ)- extremal Green function V ∗ δ (z,K, ψ) defined by the class Lδ = { u(z) ∈ psh(Cn) : u(z) 6 Cu + δ ln+ |z|, z ∈ Cn} , δ > 0. We see that the notion of regularity of points with respect to different numbers δ differ from each other. Nevertheless, we prove that if a compact set K ⊂ Cn is regular, then δ-extremal function is continuous in the whole space Cn

Sharper Bounds and Structural Results for Minimally Nonlinear 0-1 Matrices

The extremal function $ex(n, P)$ is the maximum possible number of ones in any 0-1 matrix with $n$ rows and $n$ columns that avoids $P$. A 0-1 matrix $P$ is called minimally nonlinear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every $P'$ that is contained in $P$ but not equal to $P$. Bounds on the number of ones and the number of columns in a minimally nonlinear 0-1 matrix with $k$ rows were found in (CrowdMath, 2018). In this paper, we improve the upper bound on the number of ones in a minimally nonlinear 0-1 matrix with $k$ rows from $5k-3$ to $4k-4$. As a corollary, this improves the upper bound on the number of columns in a minimally nonlinear 0-1 matrix with $k$ rows from $4k-2$ to $4k-4$. We also prove that there are not more than four ones in the top and bottom rows of a minimally nonlinear matrix and that there are not more than six ones in any other row of a minimally nonlinear matrix. Furthermore, we prove that if a minimally nonlinear 0-1 matrix has ones in the same row with exactly $d$ columns between them, then within these columns there are at most $2d-1$ rows above and $2d-1$ rows below with ones.

On the Existence of an Extremal Function in the Delsarte Extremal Problem

This paper is concerned with a Delsarte-type extremal problem. Denote by $${\mathcal {P}}(G)$$ P ( G ) the set of positive definite continuous functions on a locally compact abelian group G. We consider the function class, which was originally introduced by Gorbachev, $$\begin{aligned}&{\mathcal {G}}(W, Q)_G = \left\{ f \in {\mathcal {P}}(G) \cap L^1(G)~:\right. \\&\qquad \qquad \qquad \qquad \qquad \left. f(0) = 1, ~ {\text {supp}}{f_+} \subseteq W,~ {\text {supp}}{\widehat{f}} \subseteq Q \right\} \end{aligned}$$ G ( W , Q ) G = f ∈ P ( G ) ∩ L 1 ( G ) : f ( 0 ) = 1 , supp f + ⊆ W , supp f ^ ⊆ Q where $$W\subseteq G$$ W ⊆ G is closed and of finite Haar measure and $$Q\subseteq {\widehat{G}}$$ Q ⊆ G ^ is compact. We also consider the related Delsarte-type problem of finding the extremal quantity $$\begin{aligned} {\mathcal {D}}(W,Q)_G = \sup \left\{ \int _{G} f(g) \mathrm{d}\lambda _G(g) ~ : ~ f \in {\mathcal {G}}(W,Q)_G\right\} . \end{aligned}$$ D ( W , Q ) G = sup ∫ G f ( g ) d λ G ( g ) : f ∈ G ( W , Q ) G . The main objective of the current paper is to prove the existence of an extremal function for the Delsarte-type extremal problem $${\mathcal {D}}(W,Q)_G$$ D ( W , Q ) G . The existence of the extremal function has recently been established by Berdysheva and Révész in the most immediate case where $$G={\mathbb {R}}^d$$ G = R d . So, the novelty here is that we consider the problem in the general setting of locally compact abelian groups. In this way, our result provides a far reaching generalization of the former work of Berdysheva and Révész.

Sharp bounds for decomposing graphs into edges and triangles

For a real constant α, let $\pi _3^\alpha (G)$ be the minimum of twice the number of K2’s plus α times the number of K3’s over all edge decompositions of G into copies of K2 and K3, where Kr denotes the complete graph on r vertices. Let $\pi _3^\alpha (n)$ be the maximum of $\pi _3^\alpha (G)$ over all graphs G with n vertices. The extremal function $\pi _3^3(n)$ was first studied by Győri and Tuza (Studia Sci. Math. Hungar.22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova (Combin. Probab. Comput.28 (2019) 465–472) proved via flag algebras that $\pi _3^3(n) \le (1/2 + o(1)){n^2}$ . We extend their result by determining the exact value of $\pi _3^\alpha (n)$ and the set of extremal graphs for all α and sufficiently large n. In particular, we show for α = 3 that Kn and the complete bipartite graph ${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$ are the only possible extremal examples for large n.

Improved Adams-type inequalities and their extremals in dimension 2m

In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space [Formula: see text], where [Formula: see text] is any bounded, smooth, open subset of [Formula: see text], [Formula: see text]. Moreover, we extend this result to improved versions of Adams’ inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.

Extremal problems for convex geometric hypergraphs and ordered hypergraphs

An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and determine the order of magnitude of the extremal function for various ordered and convex geometric paths and matchings. Our results generalize earlier works of Braı–Károlyi–Valtr, Capoyleas–Pach, and Aronov–Dujmovič–Morin–Ooms-da Silveira. We also provide a new variation of the Erdős-Ko-Rado theorem in the ordered setting.

Generalized Ramsey Numbers: Forbidding Paths with Few Colors

Let $f(K_n, H, q)$ be the minimum number of colors needed to edge-color $K_n$ so that every copy of $H$ is colored with at least $q$ colors. Originally posed by Erdős and Shelah when $H$ is complete, the asymptotics of this extremal function have been extensively studied when $H$ is a complete graph or a complete balanced bipartite graph. Here we investigate this function for some other $H$, and in particular we determine the asymptotic behavior of $f(K_n, P_v, q)$ for almost all values of $v$ and $q$, where $P_v$ is a path on $v$ vertices.