It does an okay job, but there are still a lot of unanswered questions, or assumed knowledge for a lay person:
> Mathematicians want to know when such a graph is likely to have some sort of interesting structure.
Why?
> a Hamiltonian cycle, a chain of edges that passes through every vertex exactly once
> adding more edges to a graph that already contains the property will not destroy the property.
Adding one edge after the exact number of edges required to create a Hamiltonian cycle (number of edges equal to number of vertices) would appear to break the property.
> mathematicians often rely on an easier computation, one that provides a minimum possible value, or lower bound, for the threshold [of probability]
How can there be a lower bound other than zero? However small the possibility, surely given an infinite number of cases, there are infinite possibilities of a particular structure being created.
> The sunflower conjecture considers whether collections of sets can be constructed in ways that resemble the petals of a sunflower.
What does this mean?
> If Hamiltonian cycles are “spread out” nicely, that means that not too many cycles contain the same edge or subset of edges.
This seems to suggest multiple Hamiltonian cycles in a graph, contradicting the earlier definition that every vertex must be connected. I guess they meant every vertex in any particular Hamiltonian cycle.
And so on...
> Mathematicians want to know when such a graph is likely to have some sort of interesting structure.
Why?
> a Hamiltonian cycle, a chain of edges that passes through every vertex exactly once > adding more edges to a graph that already contains the property will not destroy the property.
Adding one edge after the exact number of edges required to create a Hamiltonian cycle (number of edges equal to number of vertices) would appear to break the property.
> mathematicians often rely on an easier computation, one that provides a minimum possible value, or lower bound, for the threshold [of probability]
How can there be a lower bound other than zero? However small the possibility, surely given an infinite number of cases, there are infinite possibilities of a particular structure being created.
> The sunflower conjecture considers whether collections of sets can be constructed in ways that resemble the petals of a sunflower.
What does this mean?
> If Hamiltonian cycles are “spread out” nicely, that means that not too many cycles contain the same edge or subset of edges.
This seems to suggest multiple Hamiltonian cycles in a graph, contradicting the earlier definition that every vertex must be connected. I guess they meant every vertex in any particular Hamiltonian cycle.
And so on...