A note on the stability number of an orthogonality graph

A note on orthogonality of subspaces in Euclidean geometry

We show that Euclidean geometry in suitably high dimension can be expressed as a theory of orthogonality of subspaces with fixed dimensions and fixed dimension of their meet.

A note on a stability result for the Fano plane

In this note, we adapt the Keevash-Sudakov proof of the (Turán) Stability Theorem for the Fano plane to find an explicit dependency between the parameters $\varepsilon$ and $δ$. This is useful in the solution of a multicolored version for hypergraphs of an extremal problem about edge-colorings, known as the Erdős-Rothschild problem, which may be considered for the Fano plane.

A note on stability for maximal $F$-free graphs

Popielarz, Sahasrabudhe and Snyder in 2018 proved that maximal $K_{r+1}$-free graphs with $(1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})$ edges contain a complete $r$-partite subgraph on $n-o(n)$ vertices. This was very recently extended to odd cycles in place of $K_3$ by Wang, Wang, Yang and Yuan. We further extend it to some other 3-chromatic graphs, and obtain some other stability results along the way.

A note on Bridgeland stability conditions and Catalan numbers

In this short note, we describe a problem in algebraic geometry where the solution involves Catalan numbers. More specifically, we consider the derived category of coherent sheaves on an elliptic surface, and the action of its autoequivalence group on its Bridgeland stability manifold. In solving an equation involving this group action, the generating function of Catalan numbers arises, allowing us to use asymptotic estimates of Catalan numbers to arrive at a bound for the solution set.

内部 plethysm 的稳定性性质（讲义）

Stability properties of inner plethyms (Lecture Notes)

本讲义系统阐述对称函数内部 plethysm 关于 $n$ 的稳定性，沟通顶点算子与置换矩阵特征值两种研究路径，并给出相关新结果的新证明。

A note on stability properties of powers of polymatroidal ideals

Let $I$ be a matroidal ideal of degrre $d$ of a polynomial ring $R=K[x_1,...,x_n]$, where $K$ is a field. Let astab$(I)$ and dstab$(I)$ be the smallest integer $n$ for which Ass$(I^n)$ and depth$(I^n)$ stabilize, respectively. In this paper, we show that astab$(I)=1$ if and only if dstab$(I)=1$. Moreover, we prove that if $d=3$, then ${\rm astab}(I)={\rm dstab}(I)$. Furthermore, we show that if $I$ is an almost square-free Veronese type ideal of degree $d$, then ${\rm astab}(I)={\rm dstab}(I)=\lceil\frac{n-1}{n-d}\rceil$.