There exist at least m - n linearly independent solutions to this system

There exist m - n linearly independent vectors such that every solution is a linear combination of these vectors

There exists a non-zero solution in which at least m - n variables are 0

There exists a solution in which at least n variables are non-zero

$G$ contains a complete subgraph with $k$ vertices

$G$ contains an independent set of size at least $n/k$

$G$ contains at least $k(k-1)/2$ edges

$G$ contains a vertex of degree at least $k$