Related papers: Erdos-Ko-Rado theorem for the group $\textrm{PSU}(…
Delete the edges of a Kneser graph independently of each other with some probability: for what probabilities is the independence number of this random graph equal to the independence number of the Kneser graph itself? We prove a sharp…
We use an algebraic method to prove a degree version of the celebrated Erd\H os-Ko-Rado theorem: given $n>2k$, every intersecting $k$-uniform hypergraph $H$ on $n$ vertices contains a vertex that lies on at most $\binom{n-2}{k-2}$ edges.…
We present a "modern" approach to the Erd\H{o}s-Ko-Rado theorem for Q-polynomial distance-regular graphs and apply it to the twisted Grassmann graphs discovered in 2005 by van Dam and Koolen.
A $(k,\ell)$-partition is a set partition which has $\ell$ blocks each of size $k$. Two uniform set partitions $P$ and $Q$ are said to be partially $t$-intersecting if there exist blocks $P_{i}$ in $P$ and $Q_{j}$ in $Q$ such that $\left|…
We show an inequality involving the third largest or second smallest dual eigenvalues of $Q$-polynomial association schemes of class at least three. Also we characterize dual-tight $Q$-polynomial association schemes of class three. Our…
Consider classical Kneser's graph $K(n,r)$: for two natural numbers $ r, n $ such that $r \le n / 2$, its vertices are all the subsets of $[n]=\{1,2,\ldots,n\}$ of size $r$, and two such vertices are adjacent if the corresponding subsets…
The graphs $D(k,q)$ have connected components $CD(k,q)$ giving the best known bounds on extremal problems with {\em forbidden\/} even cycles, and are denser than the well-known graphs of Lubotzky, Phillips, Sarnak and Margulis. Despite…
For a prime power $q$, let $ER_q$ denote the Erd\H{o}s-R\'enyi orthogonal polarity graph. We prove that if $q$ is an even power of an odd prime, then $\chi ( ER_{q}) \leq 2 \sqrt{q} + O ( \sqrt{q} / \log q)$. This upper bound is best…
The recent notion of $q$-deformed irrational numbers is characterized by the invariance with respect to the action of the modular group $\PSL(2,\Z)$, or equivalently under the Burau representation of the braid group~$B_3$. The theory of…
A family of sets is intersecting if no two of its members are disjoint, and has the Erd\H{o}s-Ko-Rado property (or is EKR) if each of its largest intersecting subfamilies has nonempty intersection. Denote by $\mathcal{H}_k(n,p)$ the random…
In this note, we investigate some properties of local Kneser graphs defined in [8]. In this regard, as a generalization of the Erd${\rm \ddot{o}}$s-Ko-Rado theorem, we characterize the maximum independent sets of local Kneser graphs. Next,…
The celebrated {Erd\H{o}s-Ko-Rado} Theorem states that for $n \geq 2k$ a family $\mathscr{F}$ of $k$ subsets of $[n]$ for which each pair of members of $\mathscr{F}$ have a non-empty intersection has size at most $\binom{n-1}{k-1}$ and for…
We determine the $3$-class groups of $\mathbb{Q}(\sqrt[3]{p})$ and $K=\mathbb{Q}(\sqrt[3]{p},\sqrt{-3})$ when $p\equiv 4,7\bmod 9$ is a prime and $3$ is a cubic modulo $p$. This confirms a conjecture made by Barrucand-Cohn, and proves the…
A subset of a group G of Sym(n) is intersecting if for any pair of permutations $\pi,\sigma \in G$ there is an $i$ in {1,2,...,n} such that $\pi(i) = \sigma(i)$. It has been shown, using an algebraic approach, that the largest intersecting…
Let $M_k$ be a $2n$-vertex graph with $n$ pairwise disjoint edges and let $\mathcal{H}^{(p,s)}(n)$ be the family of subsets of $V(M_n)$ that span exactly $p$ edges and $s$ isolated vertices. We prove that for $n\ge 2p+s$ this family has the…
In 2013, Ku and Wong showed that for any partitions $\mu$ and $\mu'$ of a positive integer $n$ with the same first part $u$ and the lexicographic order $\mu\triangleleft \mu'$, the eigenvalues $\xi_{\mu}$ and $\xi_{\mu'}$ of the derangement…
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…
We establish the restricted sumset analogue of the celebrated conjecture of S\'{a}rk\"{o}zy on additive decompositions of the set of nonzero squares over a finite field. More precisely, we show that if $q>13$ is an odd prime power, then the…
The well-known Erdos-Ko-Rado Theorem states that if F is a family of k-element subsets of {1,2,...,n} (n>2k-1) such that every pair of elements in F has a nonempty intersection, then |F| is at most $\binom{n-1}{k-1}$. The theorem also…
We continue our investigation of Erd\H{o}s-Ko-Rado (EKR) sets of flags in spherical buildings. In previous work, we used the theory of buildings and Iwahori-Hecke algebras to obtain upper bounds on their size. As the next step towards the…