Consider a circle $\omega$ with radius $1$. We will determine the set of all positive real numbers $t$ such that for each positive integer $n$, there exists a $\emph{good}$ collection $T$ of $n$ triangles inscribed in $\omega$, where each triangle has a perimeter greater than $t$. A $\emph{good}$ collection of triangles satisfies the following conditions:

1. Each triangle from $T$ is inscribed in $\omega$.
2. No two triangles from $T$ have a common interior point.

Since each triangle in $T$ is inscribed in a circle $\omega$ with radius $1$, the maximum possible perimeter of any such triangle is achieved when the triangle becomes an equilateral triangle. The side length $s$ of an equilateral triangle inscribed in a circle of radius $1$ is $s = \sqrt{3}$, and the perimeter $P$ of an equilateral triangle is given by:
$$P = 3s = 3\sqrt{3}.$$

For the set of all $n$ triangles to be good, their interior must not overlap. To satisfy this condition, each triangle can be made smaller by reducing the arc between successive vertices of the triangles on the circle. However, as we consider an infinitely large number of $n$ triangles, we can approach the situation where each triangle becomes a chord of the circle, and thus each triangle will have a perimeter arbitrarily close to $2\pi$, the circumference of the circle (since $2\pi \approx 6.283$).

Hence, to ensure the existence of a good collection of $n$ triangles for each positive integer $n$, the condition $t < 4$ must hold. This is because inscribed triangles with perimeters converging to $2\pi$ can be configured for any finite $n$, respecting the non-overlapping constraint. Finally:

If $t \leq 4$, then $n$ triangles of perimeter greater than $t$ can still be configured into $\omega$. Therefore, the range for $t$ allowing the existence of such collections for any $n$ is:
$$0 < t \leq 4.$$

Thus, the complete set of positive real numbers $t$ is:
$$\boxed{0 < t \leq 4}.$$

This solution verifies that for every positive integer $n$, it is feasible to construct triangles in $T$ with perimeter $> t$, with $t$ up to $4$.