Suppose you have $n$ triangles whose corners are random points on a sphere $S$ in $\mathbb{R}^3$. Viewing the triangles as built from rigid bars as edges, two triangles are linked if they cannot be separated without two edges passing through one another. A triangle that is not topologically linked with any other is loose: it could be removed without disturbing the others. In the example below of $n=15$ triangles, $11$ are linked to at least one other triangle, and $4$ are loose.
[![Tri15_4][1]][1]
$n=15$. Magenta triangles $\{1,5,10,11\}$ are loose:.
It is easy to surmise that the proportion of linked triangles approaches $1$ as $n \to \infty$:
[![TangledGraph][2]][2]
Fraction of triangles linked to at least one other triangle.
But I wonder about the largest *linked component*, the collection of all triangles linked into one "giant" component, in the sense that if you picked up one triangle all the others would follow. I wonder that when $n \to \infty$, what is the probability that this linked component includes *all* the triangles.
Q. As $n \to \infty$, what is the probability that every triangle is linked into one giant component?
My sense is that this probability is zero: Even though the probability that each triangle is linked to another approaches $1$, the probability that all triangles are linked to one another approaches $0$. This would contrast with an earlier related question, Random rings linked into one component?, whose answer was the opposite: The rings form one component as $n \to \infty$.