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Suppose we have a $(2m-1) \times (2m-1)$ matrix defined as follows: $$\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}.$$ For example, if $m=3$, the matrix is $$\begin{pmatrix}6 & 20 & 6& 0 ...

So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's ...

Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets. Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$. Can $\text{Proj}(P)$ have more than $f$ facets? ...

Definition 1: A clutter $C$ is said to have the packing property if $C$ and all of its minors satisfy the König property. where, vertex cover of $C$ is a set of vertices that have non-empty ...

This question is inspired by the discussion at this problem. Suppose I have a design consisting of a finite point set $U$ of size $|U|=m_{\emptyset}$ and a family of $n$ subsets (sometimes called ...