Conjecture If is the adjacency matrix of a -regular graph, then there is a symmetric signing of (i.e. replace some entries by ) so that the resulting matrix has all eigenvalues of magnitude at most .
Conjecture The minimum -norm of a path partition on is no more than the maximal size of an induced -colorable subgraph.
Conjecture If are invertible matrices with entries in for a prime , then there is a submatrix of so that is an AT-base.
Conjecture For every prime , there is a constant (possibly ) so that the union (as multisets) of any bases of the vector space contains an additive basis.