For any property $Q(m)$ with eventual counterexamples, the property $P(n)=$ "property $Q$ holds up to $n$" admits a largest instance, at the same number.

Conversely, for any property $P(n)$ with a largest instance, the property $Q(m)=$ "property $P$ holds somewhere above $n$" admits eventual counterexamples, starting at the same number.

The other examples are great! Meanwhile, there is a translation between this question and the eventual counterexamples question. Namely,

• For any property $Q(m)$ with eventual counterexamples, the property $P(n)=$ "property $Q$ holds up to $n$" admits a largest instance, at the same number.

• Conversely, for any property $P(n)$ with a largest instance, the property $Q(m)=$ "property $P$ holds somewhere above $n$" admits eventual counterexamples, starting at the same number.

For any property $Q(m)$ with eventual counterexamples, the property $P(n)=$ "property $Q$ holds up to $n$" admits a largest instance.

Conversely, for any property $P(n)$ with a largest instance, the property $Q(m)=$ "property $P$ holds somewhere above $n$" admits eventual counterexamples.

For any property $Q(m)$ with eventual counterexamples, the property $P(n)=$ "property $Q$ holds up to $n$" admits a largest instance, at the same number.

Conversely, for any property $P(n)$ with a largest instance, the property $Q(m)=$ "property $P$ holds somewhere above $n$" admits eventual counterexamples, starting at the same number.