Related papers: Stability and exact Turan numbers for matroids
The Erd\H{o}s discrepancy problem, now a theorem by T. Tao, asks whether every sequence with values plus or minus one has unbounded discrepancy along all homogeneous arithmetic progressions. We establish weighted variants of this problem,…
Since its formulation, Tur\'an's hypergraph problems have been among the most challenging open problems in extremal combinatorics. One of them is the following: given a $3$-uniform hypergraph $\mathcal{F}$ on $n$ vertices in which any five…
Our work is motivated by the fact that the norms of the Eulerian integers are related to the sums of form $a^2-ab+b^2$, providing a natural generalization for problems concerning products over sums or differences of integers. Let $E$ be the…
We prove an abstract Birkhoff normal form theorem for Hamiltonian partial differential equations on torus. The normal form is complete up to arbitrary finite order. The proof is based on a valid non-resonant condition and a suitable norm of…
We study the following generalization of the Tur\'an problem in sparse random graphs. Given graphs $T$ and $H$, let $\mathrm{ex}\big(G(n,p), T, H\big)$ be the random variable that counts the largest number of copies of $T$ in a subgraph of…
A classical object in hypergraph Tur\'{a}n theory is the Fano plane $\mathbb{F}$, the unique linear $3$-graph on seven vertices with seven edges. The Tur\'{a}n density and exact Tur\'{a}n number of $\mathbb{F}$, first proposed as a problem…
We consider a many-to-one variant of the stable matching problem. More concretely, we consider the variant of the stable matching problem where one side has a matroid constraint. Furthermore, we consider the situation where the preference…
In the Euclidean Bottleneck Steiner Tree problem, the input consists of a set of $n$ points in $\mathbb{R}^2$ called terminals and a parameter $k$, and the goal is to compute a Steiner tree that spans all the terminals and contains at most…
The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the…
In a finite system driven out of equilibrium by a constant external force the thermodynamic uncertainty relation (TUR) bounds the variance of the conjugate current variable by the thermodynamic cost of maintaining the non-equilibrium…
For $M$ $\omega$-categorical and stable, we investigate the growth rate of $M$, i.e. the number of orbits of $Aut(M)$ on $n$-sets, or equivalently the number of $n$-substructures of $M$ after performing quantifier elimination. We show that…
In this paper, we consider a matroid generalization of the stable matching problem. In particular, we consider the setting where preferences may contain ties. For this generalization, we propose a polynomial-time algorithm for the problem…
As a variant of the famous Tur\'an problem, we study $\mathrm{rex}(n,F)$, the maximum number of edges that an $n$-vertex regular graph can have without containing a copy of $F$. We determine $\mathrm{rex}(n,K_{r+1})$ for all pairs of…
The Zarankiewicz problem asks for an estimate on $z(m, n; s, t)$, the largest number of $1$'s in an $m \times n$ matrix with all entries $0$ or $1$ containing no $s \times t$ submatrix consisting entirely of $1$'s. We show that a classical…
We give a simple example of an $n$-tuple of orthonormal elements in $L_2$ (actually martingale differences) bounded by a fixed constant, and hence subgaussian with a fixed constant but that are Sidon only with constant $\approx \sqrt n$.…
We extend the notion of numerical stability of finite difference approximations to include hyperbolic systems that are first order in time and second order in space, such as those that appear in Numerical Relativity. By analyzing the symbol…
This is the second of two articles on the study of a particle system model that exhibits a Turing instability type effect. About the hydrodynamic quations obtained in \cite{CSL17a}, we find conditions under which Turing instability occurs…
Given graphs $H$ and $F$, $\mathrm{ex}(n,H,F)$ denotes the largest number of copies of $H$ in $F$-free $n$-vertex graphs. Let $\chi(H)<\chi(F)=r+1$. We say that $H$ is $F$-Tur\'an-stable if the following holds. For any $\varepsilon>0$ there…
We give an explicit arithmetical condition which guarantees the existence of the unstable manifold of the MacKay approximate renormalisation scheme for the breakup of invariant tori in one and a half degrees of freedom Hamiltonian systems,…
The paper investigates the stability properties of restrictions of irreducible representations of the symmetric group to the hyperoctahedral subgroup. A stability result is obtained, analogous to the classical Murnaghan theorem on the…