# Largest subsets of quadrics consisting of “nonorthogonal” vectors

Assume we have an $A$-module $M$, and a quadratic form $q : M \to A$. Recall that it means that

1) $q (a m) = a^2 q (m)$ for all $a \in A$ and $m \in M$, and

2) $B_q (x, y) := q (x + y) - q (x) - q (y)$ is a bilinear form.

Let $Q = \{ m : q (m) = 0 \}$ be the quadric corresponding to $q$.

Question: assume $A$ and $M$ are finite. Given $(M, q)$, I need to find the largest (or as large as possible) subset $S$ of $Q$ such that for any $m_1, m_2 \in S$ with $m_1 \neq m_2$, $B_q (m_1, m_2) \neq 0$.

Any pointers are welcome (known upper and lower bounds, books, papers...).

More generally, I am not fixed on $q$ itself, and one can specify finding $q$ as part of the upper and lower bounds.

• Is $M$ finite? If not, if $S$ is infinite, how should we measuer size? – Will Sawin Mar 12 '13 at 21:49
• Yes, finite. Edited the post. – CCat Mar 13 '13 at 0:04