Related papers: Cancellative hypergraphs and Steiner triple system…

We present new short proofs to both the exact and the stability results of two extremal problems. The first one is the extension of Tur\'{a}n's theorem in hypergraphs, which was firstly studied by Mubayi $\cite{MU06}$. The second one is…

We explicitly describe, in the language of four-dimensional N=1 supersymmetric field theory, what happens when the moduli of a heterotic Calabi-Yau compactification change so as to make the internal non-Abelian gauge fields…

Given an infinite reductive algebraic group $G$, we consider $G$-equivariant coherent sheaves with prescribed multiplicities, called $(G,h)$-constellations, for which two stability notions arise. The first one is analogous to the…

A {\em cross-free} set of size $m$ in a Steiner triple system $(V,{\cal{B}})$ is three pairwise disjoint $m$-element subsets $X_1,X_2,X_3\subset V$ such that no $B\in {\cal{B}}$ intersects all the three $X_i$-s. We conjecture that for every…

For $k \ge 4$, a loose $k$-cycle $C_k$ is a hypergraph with distinct edges $e_1, e_2, \ldots, e_k$ such that consecutive edges (modulo $k$) intersect in exactly one vertex and all other pairs of edges are disjoint. Our main result is that…

In the perfect tiling problem, we aim to cover the vertices of a hypergraph~$G$ with pairwise vertex-disjoint copies of a hypergraph $F$. There are three essentially necessary conditions for such a perfect tiling, which correspond to…

If $G$ is a finite group then there is an integer $M_G$ such that$,$ for $u\ge M_G$ and $u\equiv 1$ or $3$ (mod 6), there is a Steiner triple system $U$ on $u$ points for which ${\rm Aut} U \cong G. \ $ If $V$ is a Steiner triple system…

I show that anomaly cancellation conditions are sufficient to determine the two most important topological numbers relevant for Calabi-Yau compactification to six dimensions. This reflects the fact that K3 is the only non-trivial CY…

The order relations of continuous cancellative t-subnorms are discussed. First, we present some necessary and sufficient conditions along with several interesting sufficient criteria for the comparability of continuous cancellative…

Given a graph $G$ with vertices $\{v_1,\ldots,v_n\}$, we define $\mathcal{S}(G)$ to be the set of symmetric matrices $A=[a_{i,j}]$ such that for $i\ne j$ we have $a_{i,j}\ne 0$ if and only if $v_iv_j\in E(G)$. Motivated by the Graph…

For $s \ge 4$, the 3-uniform tight cycle $C^3_s$ has vertex set corresponding to $s$ distinct points on a circle and edge set given by the $s$ cyclic intervals of three consecutive points. For fixed $s \ge 4$ and $s \not\equiv 0$ (mod 3) we…

Let $A$ be a (not necessarily unital) separable non-elementary simple amenable C*-algebra whose tracial basis may not have finite covering dimension and may not be compact but satisfies certain condition (C). We show that $A$ is ${\cal…

We develop a theory for the existence of perfect matchings in hypergraphs under quite general conditions. Informally speaking, the obstructions to perfect matchings are geometric, and are of two distinct types: 'space barriers' from convex…

In this work the stability of perturbed linear time-varying systems is studied. The main features of the problem are threefold. Firstly, the time-varying dynamics is not required to be continuous but allowed to have jumps. Also the system…

We construct the minimal supersymmetric left-right theory and show that at the renormalizable level it requires the existence of an intermediate $B-L$ breaking scale. The subsequent symmetry breaking down to MSSM automatically preserves…

We study corotational wave maps from $(1+3)$-dimensional Minkowski space into the three-sphere. We establish the asymptotic stability of an explicitly known self-similar wave map under perturbations that are small in the critical Sobolev…

Steinerberger proposed a notion of curvature on graphs involving the graph distance matrix (J. Graph Theory, 2023). We show that nonnegative curvature is almost preserved under three graph operations. We characterize the distance matrix and…

The book number $b(G)$ of a graph $G$ is the maximum number of triangles sharing a common edge. A strengthening of Mantel's theorem due to Rademacher states that every $n$-vertex graph with more than $\lfloor n^2/4\rfloor$ edges contains at…

Symmetry breaking--the phenomenon in which the symmetry of a system is not inherited by its stable states--underlies pattern formation, superconductivity, and numerous other effects. Recent theoretical work has established the possibility…

A highly influential result of Nikiforov states that if an $n$-vertex graph $G$ contains at least $\gamma n^h$ copies of a fixed $h$-vertex graph $H$, then $G$ contains a blowup of $H$ of order $\Omega_{\gamma,H}(\log n)$. While the…