We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a set $S\subseteq [n]$ if every possible subset of $S$ appears as the intersection of $S$ with some element of $\mathcal{F}$ and we denote by $\text{Sh}(\mathcal{F})$ the family of sets shattered by $\mathcal{F}$. According to the Sauer-Shelah lemma we know that in general, every set system $\mathcal{F}$ shatters at least $|\mathcal{F}|$ sets and we call a set system shattering-extremal if $|\text{Sh}(\mathcal{F})|=|\mathcal{F}|$. Mészáros and Rónyai, among other things, gave an algebraic characterization of shattering-extremality, which offered the possibility to generalize the notion to general finite point sets. Here we extend the results obtained for set systems to this more general setting, and as an application, strengthen a result of Li, Zhang and Dong.
In this paper we construct a new family of representations for the quantum toroidal algebras of type $A_n$, which are $\ell$-extremal in the sense of Hernandez [24]. We construct extremal loop weight modules associated to level 0 fundamental weights $\varpi_\ell$ when $n=2r+1$ is odd and $\ell=1, r+1$ or $n$. To do it, we relate monomial realizations of level 0 extremal fundamental weight crystals with integrable representations of $\mathcal{U}_q(sl_{n+1}^{tor})$, and we introduce promotion operators for the level 0 extremal fundamental weight crystals. By specializing the quantum parameter, w…
If $J\subset I$ are two monomials ideals, we give a practical upper bound for the Stanley depth of $J/I$, which we call it the \emph{quasi-depth} of $J/I$. Also, we compute the quasi-depth of several classes of square free monomial ideals. We also study the Stanley depth of the monomial ideal associated of the independent sets of a graph.
Independent sets play a key role into the study of graphs and important problems arising in graph theory reduce to them. We define the monomial ideal of independent sets associated to a finite simple graph and describe its homological and algebraic invariants in terms of the combinatorics of the graph. We compute the minimal primary decomposition and characterize the Cohen--Macaulay ideals. Moreover, we provide a formula for computing the Betti numbers, which depends only on the coefficients of the independence polynomial of the graph.
Realizing resolutions of powers of extremal ideals
Chennai Mathematical Institute（金奈数学研究所）；Department of Mathematical Sciences, University of Texas at El Paso（德克萨斯大学埃尔帕索分校数学科学系）；Department of Mathematics & Statistics, Dalhousie University（达尔豪斯大学数学与统计系）；Department of Mathematics Texas State University（得克萨斯州立大学数学系）；Division of Computing, Analytics and Mathematics, University of Missouri-Kansas City（密苏里大学堪萨斯城分校计算、分析与数学部）
Many extremal problems for graphs have threshold graphs as their extremal examples. For instance the current authors proved that for fixed $k\ge 1$, among all graphs on $n$ vertices with $m$ edges, some threshold graph has the fewest matchings of size $k$; indeed either the lex graph or the colex graph is such an extremal example. In this paper we consider the problem of maximizing the number of matchings in the class of threshold graphs. We prove that the minimizers are what we call \emph{almost alternating threshold graphs}. We also discuss a problem with a similar flavor: which threshold gr…
This survey concerns regular graphs that are extremal with respect to the number of independent sets, and more generally, graph homomorphisms. More precisely, in the family of of $d$-regular graphs, which graph $G$ maximizes/minimizes the quantity $i(G)^{1/v(G)}$, the number of independent sets in $G$ normalized exponentially by the size of $G$? What if $i(G)$ is replaced by some other graph parameter? We review existing techniques, highlight some exciting recent developments, and discuss open problems and conjectures for future research.
The second author has introduced non-crossing tableaux, objects whose non-nesting analogues are semi-standard Young tableaux. We relate non-crossing tableaux to Gelfand-Tsetlin patterns and develop the non-crossing analogue of standard monomial theory. Leclerc and Zelevinsky's weakly separated sets are special cases of non-crossing tableaux, and we suggest that non-crossing tableaux may help illuminate the theory of weakly separated sets.
We introduced the notation of a set of prohibitions and give definitions of a complete set and a crucial word with respect to a given set of prohibitions. We consider 3 particular sets which appear in different areas of mathematics and for each of them examine the length of a crucial word. One of these sets is proved to be incomplete. The problem of determining lengths of words that are free from a set of prohibitions is shown to be NP-complete, although the related problem of whether or not a given set of prohibitions is complete is known to be effectively solvable.
Counting perfect edge dominating sets: extremal results and linear-time algorithms
Universidad Nacional de General Sarmiento（圣马丁国立大学）；Consejo Nacional de Investigaciones Científicas y Técnicas（国家科学研究委员会）；Instituto de Cálculo and Departamento de Computación, Universidad de Buenos Aires（布宜诺斯艾利斯大学计算系与计算研究所）；Instituto de Cálculo and Departamento de Matemática, Universidad de Buenos Aires（布宜诺斯艾利斯大学数学系与计算研究所）
Extremal problems involving independent sets are much studied. Two of the most important extremal problems in this context are concerned with the sharp upper bounds for the number of independent sets of fixed size and the independence number. In literature, these sharp upper bounds are derived in completely different contexts. In this paper, we show that both of these sharp upper bounds can be derived by considering Lex graphs. More exactly, they depend on two parameters of a sequence defined on them.
We extend the notion of standard pairs to the context of monomial ideals in semigroup rings. Standard pairs can be used as a data structure to encode such monomial ideals, providing an alternative to generating sets that is well suited to computing intersections, decompositions, and multiplicities. We give algorithms to compute standard pairs from generating sets and vice versa and make all of our results effective. We assume that the underlying semigroup ring is positively graded, but not necessarily normal. The lack of normality is at the root of most challenges, subtleties, and innovations …
We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a given set $S\subseteq [n]$ if $2^S={F \cap S : F \in \mathcal{F}}$. The Sauer inequality states that in general, a set system $\mathcal{F}$ shatters at least $|\mathcal{F}|$ sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly $|\mathcal{F}|$ sets. We characterize shattering extremal set systems of Vapnik-Chervonenkis dimension 1 in terms of their inclusion graphs. Also from the perspective of extremality, we relate set systems of bounded Vapnik-Chervonenkis dimens…
The main results of this paper are accessible with only basic linear algebra. Given an increasing sequence of dimensions, a flag in a vector space is an increasing sequence of subspaces with those dimensions. The set of all such flags (the flag manifold) can be projectively coordinatized using products of minors of a matrix. These products are indexed by tableaux on a Young diagram. A basis of "standard monomials" for the vector space generated by such projective coordinates over the entire flag manifold has long been known. A Schubert variety is a subset of flags specified by a permutation. L…
We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a given set $S\subseteq [n]$ if $2^S= \{F~\cap~S:~F~\in~\mathcal{F}\}$. The Sauer-Shelah lemma states that in general, a set system $\mathcal{F}$ shatters at least $|\mathcal{F}|$ sets. Here we concentrate on the case of equality. A set system is called \emph{shattering-extremal} if it shatters exactly $|\mathcal{F}|$ sets. A conjecture of Rónyai and the second author and of Litman and Moran states that if a family is shattering-extremal then one can add a set to it and the resulting family is still shattering-extremal. Here we p…
We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a given set $S\subseteq [n]$ if $2^S=\{F \cap S : F \in \mathcal{F}\}$. The Sauer inequality states that in general, a set system $\mathcal{F}$ shatters at least $|\mathcal{F}|$ sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly $|\mathcal{F}|$ sets. In this paper we characterize shattering-extremal set systems of Vapnik-Chervonenkis dimension $2$ in terms of their inclusion graphs, and as a corollary we answer an open question from \cite{VC1} about leaving out ele…
Generalising results of Erdős-Freud and Lindström, we prove that the largest Sidon subset of a bounded interval of integers is equidistributed in Bohr neighbourhoods. We establish this by showing that extremal Sidon sets are Fourier-pseudorandom, in that they have no large non-trivial Fourier coefficients. As a further application we deduce that, for any partition regular equation in five or more variables, every finite colouring of an extremal Sidon set has a monochromatic solution.
In a graph $G$, a subset of vertices is a dissociation set if it induces a subgraph with vertex degree at most 1. A maximum dissociation set is a dissociation set of maximum cardinality. The dissociation number of $G$, denoted by $ψ(G)$, is the cardinality of a maximum dissociation set of $G$. Extremal problems involving counting the number of a given type of substructure in a graph have been a hot topic of study in extremal graph theory throughout the last few decades. In this paper, we determine the maximum number of maximum dissociation sets in a tree with prescribed dissociation number and…
The main purpose of this paper is to prove the uniqueness of a graph attaining the maximum of the number of independent sets over all $k$-regular graphs on $n$ vertices for $2k|n$.
This article studies the \emph{$k$-forcing number} for oriented graphs, generalizing both the \emph{zero forcing number} for directed graphs and the $k$-forcing number for simple graphs. In particular, given a simple graph $G$, we introduce the maximum (minimum) oriented $k$-forcing number, denoted $\MOF_k(G)$ ($\mof_k(G)$), which is the largest (smallest) $k$-forcing number among all possible orientations of $G$. These new ideas are compared to known graph invariants and it is shown that, among other things, $\mof(G)$ equals the path covering number of $G$ while $\MOF_k(G)$ is greater than or…
For an interval [1,N] in the natural numbers, investigating subsets S of [1,N] such that |{(x,y) in S^2:x+y in S}|=0, known as sum-free sets, has attracted considerable attention. In this paper, we define r(S):=|{(x,y) in S^2: x+y in S}| and consider its behaviour as S ranges over the subsets of [1,N]. We obtain a comprehensive description of the spectrum of attainable r-values for the s-sets of [1,N], constructive existence results and structural characterizations for sets attaining extremal and near-extremal values.
In this note, we prove that if $A$ is a finite set of real numbers such that $|AA| = K|A|$, then for every polynomial $f \in \mathbb{R}[x,y]$ we have that $|f(A,A)| = Ω_{K,\operatorname{deg} f}(|A|^2)$, unless $f$ is of the form $f(x,y) = g(M(x,y))$ for some monomial $M$ and some univariate polynomial $g$.
For a fixed graph G, a maximal independent set is an independent set that is not a proper subset of any other independent set. P. Erdös, and independently, J. W. Moon and L. Moser, and R. E. Miller and D. E. Muller, determined the maximum number of maximal independent sets in a graph on n vertices, as well as the extremal graphs. In this paper we maximize the number of maximal independent sets of a fixed size for all graphs of order n and determine the extremal graphs. Our result generalizes the classical result.
We present an interactive framework that, given a membership test for a graph class $\mathcal{G}$ and a number $k$, finds and tests unavoidable sets for the class of graphs in $\mathcal{G}$ of path-width at most $k$. We put special emphasis on the case that $\mathcal{G}$ is the class of cubic graphs and tailor the algorithm to this case. In particular, we introduce the new concept of high-degree-first path-decompositions, which yields highly efficient pruning techniques. Using this framework we determine all extremal girth values of cubic graphs of path-width $k$ for all $k \in \{3,\dots, 10\}…
Let \{M_{r,s}\}_{0< r < p, 0< s < p'} be the irreducible Virasoro modules in the $(p,p')$-minimal series. In our previous paper, we have constructed a monomial basis of \oplus_{r=1}^{p-1}M_{r,s} in the case of $1<p'/p<2$. By `monomials' we mean vectors of the form ϕ^{(r_L,r_{L-1})}_{-n_L}...ϕ^{(r_1,r_{0})}_{-n_1} |r_0,s >, where ϕ_{-n}^{(r',r)} are the Fourier components of the (2,1)-primary field mapping M_{r,s} to M_{r',s}, and |r_0,s > is the highest weight vector of M_{r_0,s}. In this article, for all p<p' with p>2 and s=1, we describe a subset of such monomials which conjecturally forms a…
In this paper we present a novel approach in extremal set theory which may be viewed as an asymmetric version of Katona's permutation method. We use it to find more Turán numbers of hypergraphs in the Erdős--Ko--Rado range. An $(a,b)$-path $P$ of length $2k-1$ consists of $2k-1$ sets of size $r=a+b$ as follows. Take $k$ pairwise disjoint $a$-element sets $A_0, A_2, \dots, A_{2k-2}$ and other $k$ pairwise disjoint $b$-element sets $B_1, B_3, \dots, B_{2k-1}$ and order them linearly as $A_0, B_1, A_2, B_3, A_4\dots$. Define the (hyper)edges of $P_{2k-1}(a,b)$ as the sets of the form $A_i\cup B_{…
Let $A$ be a set of $n$ positive integers. We say that a subset $B$ of $A$ is a divisor of $A$, if the sum of the elements in $B$ divides the sum of the elements in $A$. We are interested in the following extremal problem. For each $n$, what is the maximum number of divisors a set of $n$ positive integers can have? We determine this function exactly for all values of $n$. Moreover, for each $n$ we characterize all sets that achieve the maximum. We also prove results for the $k$-subset analogue of our problem. For this variant, we determine the function exactly in the special case that $n=2k$. …
A graphical design is a proper subset of vertices of a graph on which many eigenfunctions of the Laplacian operator have mean value zero. In this paper, we show that extremal independent sets make extremal graphical designs, that is, a design on which the maximum possible number of eigenfunctions have mean value zero. We then provide examples of such graphs and sets, which arise naturally in extremal combinatorics. We also show that sets which realize the isoperimetric constant of a graph make extremal graphical designs, and provide examples for them as well. We investigate the behavior of gra…
We study extremal problems about sets of integers that do not contain sumsets with summands of prescribed size. We analyse both finite sets and infinite sequences. We also study the connections of these problems with extremal problems of graphs and hypergraphs.
We study toric degenerations of opposite Schubert and Richardson varieties inside degenerations of Grassmannians and flag varieties. These degenerations are parametrized by matching fields in the sense of Sturmfels and Zelevinsky. We construct so-called restricted matching field ideals whose generating sets are understood combinatorially through tableaux. We determine when these ideals are toric and coincide with Gröbner degenerations of Richardson varieties using the well established standard monomial theory for Grassmannians and flag varieties.
The independent set reconfiguration problem asks whether one can transform one given independent set of a graph into another, by changing vertices one by one in such a way the intermediate sets remain independent. Extremal problems on independent sets are widely studied: for example, it is well known that an $n$-vertex graph has at most $3^{n/3}$ maximum independent sets (and this is tight). This paper investigates the asymptotic behavior of maximum possible length of a shortest reconfiguration sequence for independent sets of size $k$ among all $n$-vertex graphs. We give a tight bound for $k=…
Let $G$ be a finite abelian group. The critical number ${\rm cr}(G)$ of $G$ is the least positive integer $\ell$ such that every subset $A\subseteq G\setminus\{0\}$ of cardinality at least $\ell$ spans $G$, i.e., every element of $G$ can be written as a nonempty sum of distinct elements of $A$. The exact values of the critical number have been completely determined recently for all finite abelian groups. The structure of these sets of cardinality ${\rm cr}(G)-1$ which fail to span $G$ has also been characterized except for the case that $|G|$ is an even number and the case that $|G|=pq$ with $…
In this paper we find monomial bases for the integer cohomology rings of compact wonderful models of toric arrangements. In the description of the monomials various combinatorial objects come into play: building sets, nested sets, and the fan of a suitable toric variety. We provide some examples computed via a SageMath program and then we focus on the case of the toric arrangements associated with root systems of type A. Here the combinatorial description of our basis offers a geometrical point of view on the relation between some Eulerian statistics on the symmetric group.
Let $m(n)$ denote the maximum size of a family of subsets which does not contain two disjoint sets along with their union. In 1968 Kleitman proved that $m(n) = {n\choose m+1}+\ldots +{n\choose 2m+1}$ if $n=3m+1$. Confirming the conjecture of Kleitman, we establish the same equality for the cases $n=3m$ and $n=3m+2$, and also determine all extremal families. Unlike the case $n=3m+1$, the extremal families are not unique. This is a plausible reason behind the relative difficulty of our proofs. We completely settle the case of several families as well.
A directed graph is oriented if it can be obtained by orienting the edges of a simple, undirected graph. For an oriented graph $G$, let $β(G)$ denote the size of a minimum feedback arc set, a smallest subset of edges whose deletion leaves an acyclic subgraph. A simple consequence of a result of Berger and Shor is that any oriented graph $G$ with $m$ edges satisfies $β(G) = m/2 - Ω(m^{3/4})$. We observe that if an oriented graph $G$ has a fixed forbidden subgraph $B$, the upper bound of $β(G) = m/2 - Ω(m^{3/4})$ is best possible as a function of the number of edges if $B$ is not bipartite, but …
We determine the $166\,104$ extremal monomials of the discriminant of a quaternary cubic form. These are in bijection with $D$-equivalence classes of regular triangulations of the $3$-dilated tetrahedron. We describe how to compute these triangulations and their $D$-equivalence classes in order to arrive at our main result. The computation poses several challenges, such as dealing with the sheer amount of triangulations effectively, as well as devising a suitably fast algorithm for computation of a $D$-equivalence class.
We prove that if $A \subseteq [X, 2X]$ and $B \subseteq [Y, 2Y]$ are sets of integers such that $\gcd(a,b) \geq D$ for at least $δ|A||B|$ pairs $(a,b) \in A \times B$ then $|A||B| \ll_{\varepsilon} δ^{-2 - \varepsilon} XY/D^2$. This is a new result even when $δ= 1$. The proof uses ideas of Koukoulopoulos and Maynard and some additional combinatorial arguments.
