CyberSec Research

We study asymptotics of the $d=4$, $\mathcal{N}=1$ superconformal index for toric quiver gauge theories. Using graph-theoretic and algebraic factorization techniques, we obtain a cycle expansion for the large-$N$ index in terms of the $R$-charge-weighted adjacency matrix. Applying saddle-point techniques at the on-shell $R$-charges, we determine the asymptotic degeneracy in the univariate specialization for $\hat{A}_{m}$, and along the main diagonal for the bivariate index for $\mathcal{N}=4$ and $\hat{A}_{3}$. In these cases we find $\ln |c_{n}| \sim \gamma n^{\frac{1}{2}}+ \beta \ln n + \alpha$ (Hardy-Ramanujan type). We also identify polynomial growth for $dP3$, $Y^{3,3}$ and $Y^{p,0}$, and give numerical evidence for $\gamma$ in further $Y^{p,p}$ examples. Finally, we generalize Murthy's giant graviton expansion via the Hubbard-Stratonovich transformation and Borodin-Okounkov formula to multi-matrix models relevant for quivers.

Erd\H{o}s asked whether for any $n$-vertex graph $G$, the parameter $p^*(G)=\min \sum_{i\ge 1} (|V(G_i)|-1)$ is at most $\lfloor n^2/4\rfloor$, where the minimum is taken over all edge decompositions of $G$ into edge-disjoint cliques $G_i$. In a restricted case (also conjectured independently by Erd\H{o}s), Gy\H{o}ri and Keszegh [Combinatorica, 37(6) (2017), 1113--1124] proved that $p^*(G)\leq \lfloor n^2/4\rfloor$ for all $K_4$-free graphs $G$. Motivated by their proof approach, they conjectured that for any $n$-vertex $K_4$-free graph $G$ with $e$ edges, and any greedy partition $P$ of $G$ of size $r$, the number of triangles in $G$ is at least $r(e-r(n-r))$. If true, this would imply a stronger bound on $p^*(G)$. In this paper, we disprove their conjecture by constructing infinitely many counterexamples with arbitrarily large gap. We further establish a corrected tight lower bound on the number of triangles in such graphs, which would recover the conjectured bound once some small counterexamples we identify are excluded.

In this paper, we study a multicolor variant of Erd\H{o}s--Rogers functions. Let $f_{\alpha_s; K_{i_1}, \cdots, K_{i_t}}(n)$ be the largest integer $m$ such that there is always an induced $K_s$-free subgraph of size $m$ in every $n$-vertex graph with a $t$-edge-coloring in which the edges with the $j$-th color induce no copy of $K_{i_j}$. We establish both upper and lower bounds for this multicolor version. Specifically, we show that $f_{\alpha_5; K_3, K_3}(n) = n^{1/2+o(1)}$, $\Omega(n^{5/11}) \le f_{\alpha_5; K_3, K_3, K_3}(n) \le n^{1/2+o(1)}$, and $\Omega(n^{20/61}) \le f_{\alpha_5; K_3, K_3, K_3, K_3}(n) \le n^{1/3+o(1)}$.

We provide two new proofs of the infinitude of prime numbers, using the additive Ramsey-theoretic result known as Folkman's theorem (alternatively, one can think of these proofs as using Hindman's theorem). This adds to the existing literature deriving the infinitude of primes from Ramsey-type theorems.

Let $R$ be a finite ring with identity. The clean graph $Cl(R)$ of a ring $R$ is a graph whose vertices are pairs $(e, u)$, where $e$ is an idempotent element and $u$ is a unit of $R$. Two distinct vertices $(e, u)$ and $(f, v)$ are adjacent if and only if $ef = fe = 0$ or $uv = vu = 1$. The graph $Cl_2(R)$ is the induced subgraph of $Cl(R)$ induced by the set $\{(e, u): e \text{ is a nonzero idempotent and } u \text{ is a unit of } R\}$. In this study, we present properties that arise from the isomorphism of two clean graphs and conditions under which two clean graphs over direct product rings are isomorphic. We also examine the structure of the clean graph over the ring $M_2(\mathbb{Z}_p)$ through their $Cl_2$ graph.

In 2006, Collins and Trenk obtained a general sharp upper bound for the distinguishing chromatic number of a connected graph. Inspired by Catlin's combinatorial techniques from 1978, we establish improved upper bounds for classes of connected graphs that have only small complete bigraphs as induced subgraphs. In this framework, we also consider the list-distinguishing chromatic number of such graphs. We apply Menger's theorem to demonstrate applications of our main result for graphs whose constructions are based on Paley graphs, Cayley graphs on Dihedral groups, and circulant Cayley graphs.

Let $I$ be a polymatroidal ideal. In this paper, we study the asymptotic behavior of the homological shift ideals of powers of polymatroidal ideals. We prove that the first homological shift algebra $\text{HS}_1(\mathcal{R}(I))$ of $I$ is generated in degree one as a module over the Rees algebra $\mathcal{R}(I)$ of $I$. We conjecture that the $i$th homological shift algebra $\text{HS}_i(\mathcal{R}(I))$ of $I$ is generated in degrees $\le i$, and we confirm it in many significant cases. We show that $I$ has the $1$st homological strong persistence property, and we conjecture that the sequence $\{\text{Ass}\,\text{HS}_i(I^k)\}_{k>0}$ of associated primes of $\text{HS}_i(I^k)$ becomes an increasing chain for $k\ge i$. This conjecture is established when $i=1$ and for many families of polymatroidal ideals. Finally, we explore componentwise polymatroidal ideals, and we prove that $\text{HS}_1(I)$ is again componentwise polymatroidal, if $I$ is componentwise polymatroidal.

In 1917, Huntington and Kline, followed by Huntington in 1924, studied systems of axioms for ternary relations aiming to capture the concepts of linear order (called betwenness) and cycle order, respectively. Among many other properties, they proved there are several independent set of axioms defining either linear order or cycle order. In this work, we consider systems arising from the four common axioms of linear order and cycle order, together with a new axiom F, stating that if a tuple abc is in the relation, then either cba or bca is in the relation as well. Although, at first glace this allows for a much richer type of systems, we prove that these are either of linear order or cycle order type. Our main result is a complete characterization of the finite systems satisfying the set of axioms {B, C, D, F, 2}, where B, C, D and 2 are axioms presented by Huntington. Unlike what happens in the previous situation, with this modification we obtain a larger family of systems which we characterize in terms of digraphs.

Motivated by the analogy with the Coxeter complex on one side, and parking functions on the other side, we study the poset of parabolic cosets in a finite Coxeter group. We show that this poset is Cohen-Macaulay, and get an explicit formula for the character of its (unique) nonzero homology group in terms of the M\"obius function of the intersection lattice. This homology character becomes a positive element of the parabolic Burnside ring (in its natural basis) after tensoring with the sign character. The coefficients of this character essentially encode the colored $h$-vector of the positive chamber complex (following Bastidas, Hohlweg, and Saliola, this complex is defined by taking Weyl chambers that lie on the positive side of a generic hyperplane). Roughly speaking, tensoring by the sign character on one side corresponds to the transformation going from the $f$-vector to the $h$-vector on the other side.

We define an all-$k$-isolating set of a graph to be a set $S$ of vertices such that, if one removes $S$ and all its neighbors, then no component in what remains has order $k$ or more. The case $k=1$ corresponds to a dominating set and the case $k=2$ corresponds to what Caro and Hansberg called an isolating set. We show that every tree of order $n \neq k$ contains an all-$k$-isolating set $S$ of size at most $n/(k+1)$, and moreover, the set $S$ can be chosen to be an independent set. This extends previous bounds on variations of isolation, while improving a result of Luttrell et al., who called the associated parameter the $k$-neighbor component order connectivity. We also characterize the trees where this bound is achieved. Further, we show that for~$k\le 5$, apart from one exception every tree with $n\neq k$ contains $k+1$ disjoint independent all-$k$-isolating sets.

The $l$-hypermaps, $l\ge2$, which generalize (dual of) ribbon graphs ($l=2$ case), are interesting enumerative objects. In this paper, based on a theorem of Carlet--van de Leur--Posthuma--Shadrin and the matrix-resolvent method, we derive an explicit formula for $k$-point generating series of enumeration of $l$-hypermaps, which generalizes the one obtained in [30] for the $l=2$ case. We also generalize a theorem of Dubrovin [29].

The class of the fine moduli space of stable $n$-pointed curves of genus zero, $\overline{\mathcal{M}_{0,n}}$, in the Grothendieck ring of varieties encodes its Poincar\'e polynomial. Aluffi-Chen-Marcolli conjecture that the Grothendieck class of $\overline{\mathcal{M}_{0,n}}$ is real-rooted (and hence ultra-log-concave), and they proved an asymptotic ultra-log-concavity result for these polynomials. We build upon their work, by providing effectively computable bounds for the error term in their asymptotic formula for $\mathrm{rk}\, H^{2l}(\overline{\mathcal{M}_{0,n}})$. As a consequence, we prove that in the range $l \le \frac{n}{10\log n}$, the ultra-log-concavity inequality \[\left(\frac{\mathrm{rk}\, H^{2(l-1)}(\overline{\mathcal{M}_{0,n}})}{\binom{n-3}{l-1}}\right)^2 \ge \frac{\mathrm{rk}\, H^{2(l-2)}(\overline{\mathcal{M}_{0,n}})\mathrm{rk}\, H^{2l}(\overline{\mathcal{M}_{0,n}})}{\binom{n-3}{l-2}\binom{n-3}{l}} \] holds for $n$ sufficiently large.

Let $G=(V, E)$ be a simple undirected graph. A closed neighbourhood of an edge $e=uv$ between two vertices $u$ and $v$ of $G$, denoted by $N_G[e]$, is the set of vertices in the neighbourhood of $u$ and $v$ including $\{u,v\}$. A subset $L$ of $V$ is said to be liar's vertex-edge dominating set if $(i)$ for every edge $e\in E$, $|N_G[e]\cap L|\geq 2$ and $(ii)$ for every pair of distinct edges $e,e'$, $|(N_G[e]\cup N_G[e'])\cap L|\geq 3$. The minimum liar's vertex-edge domination problem is to find the liar's vertex-edge dominating set of minimum cardinality. In this article, we show that the liar's vertex-edge domination problem is NP-complete in unit disk graphs, and we design a polynomial time approximation scheme(PTAS) for the minimum liar's vertex-edge domination problem in unit disk graphs.

We establish dimensional thresholds for dot product sets associated with compact subsets of translated paraboloids. Specifically, we prove that when the dimension of such a subset exceeds $ \frac{5}{4} = \frac{3}{2} - \frac{1}{4} $ in $\mathbb{R}^3$, and $ \frac{d}{2} - \frac{1}{4} - \frac{1}{8d - 4} $ in $\mathbb{R}^d$ for $d\geq 4$, its dot product set has positive Lebesgue measure. This result demonstrates that if a compact set in $ \mathbb{R}^d $ exhibits a paraboloidal structure, then the usual dimensional barrier of $ \frac{d}{2} $ for dot product sets can be lowered for $ d \geq 3 $. Our work serves as the continuous counterpart of a paper by Che-Jui Chang, Ali Mohammadi, Thang Pham, and Chun-Yen Shen, which examines the finite field setting with partial reliance on the extension conjecture. The key idea, closely following their paper, is to reformulate the dot product set on the paraboloid as a variant of a distance set. This reformulation allows us to leverage state-of-the-art results from the pinned distance problem, as established by Larry Guth, Alex Iosevich, Yumeng Ou, and Hong Wang for $ d = 2 $, and by Xiumin Du, Yumeng Ou, Kevin Ren, and Ruixiang Zhang for higher dimensions. Finally, we present explicit constructions and existence proofs that highlight the sharpness of our results.

In "Meromorphic Functions and Analytic Curves", H. and F. J. Weyl identified an intriguing connection between holomorphic curves and their associated curves, which they referred to as the "peculiar relation". In this paper, we present a generalization of the Weyl peculiar relation and investigate a combinatorial structure underlying the Weyl--Ahlfors theory via standard Young tableaux. We also provide an alternative proof of the Second Main Theorem from the viewpoint of comparing the order functions $iT_p$ and $T_i\{\mathbf{X}^{(p)}\}$.

For a natural number $k>1$, let $f_k(n)$ denote the number of distinct representations of a natural number $n$ of the form $p^k+q^k$ for primes $p,q$. We prove that, for all $k>1$, $$\limsup_{n\to\infty}f_k(n)=\infty.$$ This positively answers a conjecture of Erdos, which asks if there are natural numbers $n$ with arbitrarily many distinct representations of the form $p_1^k+p_2^k+\dots+p_k^k$ for primes $p_1,p_2,\dots,p_k$.

We revisit classic balancing problems for linear extensions of a partially ordered set $P$, proving results that go far beyond many of the best earlier results on this topic. For example, with $p(x\prec y)$ the probability that $x$ precedes $y$ in a uniform linear extension, $\delta_{xy} = \min\{p(x \prec y), p(y \prec x)\}$, and $\delta(P)=\max \delta_{xy}$, we show that $\delta(P)$ tends to $1/2$ as $n := |P| \to\infty$ if $P$ has width $\Omega(n)$ or $\omega(\log(n))$ minimal elements, and is at least $1/e-o(1)$ if $P$ has width $\omega(\sqrt{n})$ or height $o(n)$. Motivated by both consequences for balance problems and intrinsic interest, we also consider several old and new parameters associated with $P$. Here, in addition to balance, we study relations between the parameters and suggest various questions that are thought to be worthy of further investigation.

For each finite Coxeter group $W$ and each standard Coxeter element of $W$, we construct a triangulation of the $W$-permutahedron. For particular realizations of the $W$-permutahedron, we show that this is a regular triangulation induced by a height function coming from the theory of total linear stability for Dynkin quivers. We also explore several notable combinatorial properties of these triangulations that relate the Bruhat order, the noncrossing partition lattice, and Cambrian congruences. Each triangulation gives an explicit mechanism for relating two different presentations of the corresponding braid group (the standard Artin presentation and Bessis's dual presentation). This is a step toward uniformly proving conjectural simple, explicit, and type-uniform presentations for the corresponding pure braid group.