6 problems

For primes $p$ and positive integers $n$, let $\mathrm{QGS}(n,p)$ be the quadratic Green–Sanders example, and let $\mathrm{VC}_2$-dimension denote the paper's second-order VC-dimen…

Let $\mathrm{XOP}_2$ be a prospective, as-yet-undefined generalization of the order property, and write $\mathrm{NXOP}_2$ for its negation. Let a property be quadratically atomic i…

Let $\mathcal{P}$ be an elementary $p$-group property, and let $\mathrm{NFOP}_2$ mean that it has no functional order property of order two. A property is quadratically atomic when…

Let an elementary $p$-group property (epGP) be a property of elementary abelian $p$-groups, and let $\mathrm{NIP}_2$ denote the stated second-order independence-property condition.…

Let $\mathscr{A}$ be a set of positive integers satisfying … for all sufficiently large primes $p$. Symmetric inverse large sieve conjecture. Either there is a rational quadratic…

Let $X_0\in\mathbb{N}$, let $\rho>0$, and let $X\in\mathbb{N}$ be sufficiently large in terms of $X_0$ and $\rho$. Suppose that $A,B\subset[X]$ satisfy … for all…