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Let $n,k$ be positive integers. What is the smallest value of $N$ such that for any $N$ vectors (may be repeated) in $(\mathbb Z/(n))^k$, one can pick $n$ vectors whose sum is $0$? My guess is $N=2^ …

Let $\Delta$ be a flag complex on $n$ vertices. Let $r$ be the smallest size of the facets of $\Delta$. Suppose that $2r>n$. Must $\Delta$ be acyclic?

Let $A$ be a set of vectors in $\mathbb Z^d$ who $\mathbb R$-span is the whole $\mathbb R^d$. Let $s_i(A)$ denote the size of $A+A+\dots A$ ($i$ times). I am interested in the following: Question …

This question is inspired by this one. Let $P \subset \mathbb R_{+}^n$ be a convex polyhedron whose complement in $\mathbb R_{+}^n$ has finite volume. Let $Int(P) = P \cap \mathbb N^n$. (For motivatio …

There is an interpretation of your inequality using simplicial complex and $h$-vector. Namely, let $\Delta$ be the set of all $P-S$ with $S$ in your set. Then $\Delta$ is a collection closed under inc …

Let $G$ be a finite, simple, undirected, connected graph. Suppose that $G$ has maximal degree $d$ and the complement $G^c$ has no induced cycles of lengths $i$, for $4 \leq i \leq l$. My question is: …

Occasionally, but more frequently lately, I would like to perform some hard computations. As an example, yesterday the following question came up: What is the projective dimension of the edge idea …

Interestingly enough, there have been quite a few recent attempts at answering your question. First, some relevant background. The diameter that you defined is known as the diameter of the dual graph …

That Def 1 and Def 2 are equivalent is a well-known Conjecture, still open as far as I know. Curiously, you can translate the whole conjecture to the language of commutative algebra, see for example p …

Think of the hypergraph as a simplicial complex $\Delta$, with the facets being the hyperedges. Consider property (*) as: 1) The $i$-skeleton of $\Delta$ is full for $0\leq i\leq k-2$ and 2) $\ …

Let $a_1,\cdots, a_n$ be integers such that $a_i\geq 2$ for all $i$ and $k>0$ another integer. I am interested in whether there exist integers $x_1,\cdots, x_n$ with $0<x_i<a_i$ satisfying: $$ \fra …

Let $S \subset \mathbb R^d$ be finite set of points. We say that $S$ is $2$-covered if $S$ lies in a union $V_1\cup V_2$ of affine subspaces such that $\dim(V_1)+\dim (V_2)\leq d-1$. For example, if $ …

Let $[n]$ denote the set $\{0,1,...,n\}$. A subset $S\subseteq [n]$ is said to have complete double if $S+S=[2n]$. Let $m(n)$ be the smallest size of a subset of $[n]$ with complete double. My questio …

Let $T$ be a triangulation of sphere. We say that $T$ is $k$-colorable if the triangles of $T$ can be assigned with $k$ colors such that any two triangles with a common edge have different colors. I a …