Related papers: Flag Algebras: A First Glance
Razborov's flag algebra forms a powerful framework for deriving asymptotic inequalities between induced subgraph densities, underpinning many advances in extremal graph theory. This survey introduces flag algebra to computer scientists…
These are some brief notes on the translation from Razborov's recently-developed notion of flag algebra into the lexicon of functions and measures on certain abstract Cantor spaces (totally disconnected compact metric spaces).
Tur\'an problems in extremal combinatorics ask to find asymptotic bounds on the edge densities of graphs and hypergraphs that avoid specified subgraphs. The theory of flag algebras proposed by Razborov provides powerful methods based on…
We introduce FlagAlgebraToolbox, an extension of SageMath capable of automating flag algebra calculations and optimizations. FlagAlgebraToolbox has a simple interface, can handle a wide range of combinatorial theories, can numerically…
The theory of limits of discrete combinatorial objects has been thriving for the last decade or so. The syntactic, algebraic approach to the subject is popularly known as "flag algebras", while the semantic, geometric one is often…
In this paper, we prove several new Tur\'an density results for 3-graphs with independent neighbourhoods. We show: \pi(K_4^-, C_5, F_{3,2})=12/49, \pi(K_4^-, F_{3,2})=5/18, and \pi(J_4, F_{3,2})=\pi(J_5, F_{3,2})=3/8, where J_t is the…
Motivated by the Caccetta-Haggkvist Conjecture, we prove that every digraph on n vertices with minimum outdegree 0.3465n contains an oriented triangle. This improves the bound of 0.3532n of Hamburger, Haxell and Kostochka. The main new tool…
The inducibility of a graph represents its maximum density as an induced subgraph over all possible sequences of graphs of size growing to infinity. This invariant of graphs has been extensively studied since its introduction in $1975$ by…
If K is an odd-dimensional flag closed manifold, flag generalized homology sphere or a more general flag weak pseudomanifold with sufficiently many vertices, then the maximal number of edges in K is achieved by the balanced join of cycles.…
Refining a basic result of Alexander, we show that two flag simplicial complexes are piecewise linearly homeomorphic if and only if they can be connected by a sequence of flag complexes, each obtained from the previous one by either an edge…
We establish an explicit combinatorial/homological characterization of supports for linear degenerations of flag varieties. For such purpose, we introduce the concept of an excessive multisegment. It provides a new class of combinatorial…
One of the earliest results in extremal graph theory, Mantel's theorem, states that the maximum number of edges in a triangle-free graph $G$ on $n$ vertices is $\lfloor n^2/4 \rfloor$. We investigate how this extremal bound is affected when…
This is an exposition of some recent developments related to the object in the title, particularly the combinatorial computation of the (genus 0) Gromov-Witten invariants of the flag manifold and the quadratic algebra approach. The notes…
We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep…
Canonical algebras, introduced by C.M. Ringel in 1984, play an important role in the representation theory of finite dimensional algebras. They are equipped with a large contact surface to many further mathematical subjects like function…
Galois orders, introduced in 2010 by V. Futorny and S. Ovsienko, form a class of associative algebras that contain many important examples, such as the enveloping algebra of $\mathfrak{gl}_n$ (as well as its quantum deformation),…
Extremal Graph Theory is a very deep and wide area of modern combinatorics. It is very fast developing, and in this long but relatively short survey we select some of those results which either we feel very important in this field or which…
We prove that the number of multigraphs with vertex set $\{1, \ldots, n\}$ such that every four vertices span at most nine edges is $a^{n^2 + o(n^2)}$ where $a$ is transcendental (assuming Schanuel's conjecture from number theory). This is…
Grassmann and flag varieties lead many lives in pure and applied mathematics. Here we focus on the algebraic complexity of solving various problems in linear algebra and statistics as optimization problems over these varieties. The measure…
Let $\mathfrak{g}$ be a semisimple complex Lie algebra of finite dimension and $\mathfrak{h}$ be a semisimple subalgebra. We present an approach to find the branching rules for the pair $\mathfrak{g}\supset\mathfrak{h}$. According to an…