Title: Maximum Erdős-Ko-Rado sets of chambers and their antidesigns in vector-spaces of even dimension Show Chinese title

Title: 偶维向量空间中房间的最大Erdős-Ko-Rado集及其反设计

Abstract: A chamber of the vector space $\mathbb{F}_q^n$ is a set $\{S_1,\dots,S_{n-1}\}$ of subspaces of $\mathbb{F}_q^n$ where $S_1\subset S_2\subset \dotso \subset S_{n-1}$ and $\dim(S_i)=i$ for $i=1,\dots,n-1$. By $\Gamma_n(q)$ we denote the graph whose vertices are the chambers of $\mathbb{F}_q^n$ with two chambers $C_1=\{S_1,\dots,S_{n-1}\}$ and $C_2=\{T_1,\dots,T_{n-1}\}$ adjacent in $\Gamma_n(q)$, if $S_i\cap T_{n-i}=\{0\}$ for $i=1,\dots,n-1$. The Erd\H{o}s-Ko-Rado problem on chambers is equivalent to determining the structure of independent sets of $\Gamma_n(q)$. The independence number of this graph was determined in [7] for $n$ even and given a subspace $P$ of dimension one, the set of all chambers whose subspaces of dimension $\frac n2$ contain $P$ attains the bound. The dual example of course also attains the bound. It remained open in [7] whether or not these are all maximum independent sets. Using a description from [6] of the eigenspace for the smallest eigenvalue of this graph, we prove an Erd\H{o}s-Ko-Rado theorem on chambers of $\mathbb{F}_q^n$ for sufficiently large $q$, giving an affirmative answer for n even. Show Chinese abstract

Abstract: A chamber of the vector space $\mathbb{F}_q^n$ is a set $\{S_1,\dots,S_{n-1}\}$ of subspaces of $\mathbb{F}_q^n$ where $S_1\subset S_2\subset \dotso \subset S_{n-1}$ and $\dim(S_i)=i$ for $i=1,\dots,n-1$. By $\Gamma_n(q)$ we denote the graph whose vertices are the chambers of $\mathbb{F}_q^n$ with two chambers $C_1=\{S_1,\dots,S_{n-1}\}$ and $C_2=\{T_1,\dots,T_{n-1}\}$ adjacent in $\Gamma_n(q)$, if $S_i\cap T_{n-i}=\{0\}$ for $i=1,\dots,n-1$. 埃德罗斯-科-拉多关于房间的问题等价于确定独立集的结构的$\Gamma_n(q)$。这个图的独立数在 [7] 中被确定，对于$n$偶数和给定一个维数为一的子空间$P$，所有包含$P$的维数为$\frac n2$的房间的集合达到了该界限。当然，对偶例子也达到了该界限。在 [7] 中仍然未解决的是，这些是否是所有最大独立集。利用 [6] 中对该图最小特征值的特征空间的描述，我们证明了关于$\mathbb{F}_q^n$的房间的埃德罗斯-科-拉多定理，对于足够大的$q$给出了 n 为偶数的肯定答案。