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A definition of "separator" is the following: Let $G$ is an $n$-vertex graph, then $S\subseteq V(G)$ is a separator if there is a partition $V=A\cup B\cup S$ such that $|A|,|B|\le 2n/3$ and there is n …

Given a Young diagram $Y$, for each row $R$ choose a permutation $\sigma_R$ of $\{1,\dots, |R|\}$, where $|R|$ is the size of row $R$. Let $\sigma_R(i)$ be the “row coordinate” of the $i$th cell in ro …

Konig's theorem in graph theory says that for a bipartite graph $G$, the size of maximum matching in $G$ is equal to the size of minimum vertex cover of $G$. Typically, one of the proofs is to constru …

For a prime $p$, fix two bases $U=\{v_1,\dots,v_n\}$ and $W=\{w_1,\dots,w_n\}$ of the vector space $V=\mathbf{F}_p^n$. We may assume $U$ is the standard basis without loss of generality. For $s_1,\dot …

Let $C$ be a simplicial complex on a finite set $V$: that means $C$ is a collection of subsets of $V$ such that if $\sigma\in C$ and $\tau\subseteq \sigma$, then $\tau\in C$. The Alexander dual $D(C)$ …

The connectivity $\eta(X)$ of a simplicial complex $X$ is defined as the $$1+\min_j\{j \mid \tilde{H}_j(X)\neq 0\}.$$ If no such $j$ exists, then $\eta(X):=\infty$. (See here for this definition, whic …

Given a complete graph $K_n$, and if we know there are $t$ $K_s$ ($s\ge 2$) in it, what can we say about the possible number $a$ of vertices and the number $b$ edges to form these $t$ cliques? We can …

Do there exist constants $d>0$, $0<c<1$, $\delta>0$ so that for all large $n$, there exists a graph $H$ satisfying $$e_H\ge dn^2,$$ and then no matter how we remove some edges from $H$ to get an $n$-v …

Let $n$ be some parameter tending to infinity. I am wondering does there exists some kind of graphs $G$ on vertex-set $[n]$ with maximum degree less than $D$, so that $D\ge n/w_1(n)$, $e_G$, the nu …

Let $k$ be an even number. For a $k$-dimensional cube (http://mathworld.wolfram.com/HypercubeGraph.html) $Q_k$, let $G$ be a subgraph of $Q_k$ with $2^{k-1}+s$ vertices, for $1\le s\le 2^{k-1}-1$. I a …

I'm looking for a tight bound for Turan number $ex_2(n,K_{1,t,t})$, where $K_{1,t,t}$ is the complete 3-partite graph with parts of size 1, t, and t. The motivation is that we now $ex_2(n,K_{t,t})=O( …

How to prove this identity? $ \sum_{n\ge 0} \frac{x^{n^2}}{(1-x)(1-x^2)\cdots(1-x^n)}= \frac{1}{\prod_{k \ge 0}(1-x^{5k+1})(1-x^{5k+4})}$ I will appreciate it a lot if a solution using method involvin …