  Let $q,n \geq 1$ be integers, $[q]=\{1,\ldots, q\}$, and $\mathbb F$ be a
field with $|\mathbb F|\geq q$. The set of increasing sequences $$
I(n,q)=\{(f_1,f_2, \dots, f_n) \in [q]^n:~ f_1\leq f_2\leq\cdots \leq f_n \} $$
can be mapped via an injective map $i: [q]\rightarrow \mathbb F $ into a subset
$J(n,q)$ of the affine space ${\mathbb F}^n$. We describe reduced Gr\"obner
bases, standard monomials and Hilbert function of the ideal of polynomials
vanishing on $J(n,q)$.
  As applications we give an interpolation basis for $J(n,q)$, and lower bounds
for the size of increasing Kakeya sets, increasing Nikodym sets, and for the
size of affine hyperplane covers of $J(n,q)$.
