My question is about the end of the proof of Theorem 1 in (Komlós, Szemerédi (1983)), more precisely the arguments in Subsection 2.3. Let me state the beautiful theorem I am trying to understand in my own words:

**Theorem 1.** *With the vertex set ${1, 2, dots, n}$ and edges drawn independently with probability
$$frac{frac{1}{2}nln(n) + frac{1}{2}nln(ln(n) + nc_n}{binom{n}{2}}$$
for a real sequence $(c_n)$ converging to $cinmathbb{R}cup{-infty, infty}$ the probability of the event that the graph has a Hamiltonian cycle converges to the limit distribution $e^{-e^{-c}}$ (extended continuously to $0$ and $infty$ for $c=mpinfty$, respectively) for $ntoinfty$.*

The limit distribution is the one for graphs with minimal degree $2$, so this necessary condition for being Hamiltonian is almost sufficient. In the proof a series of additional graph theoretical conditions is defined, which force the graph to be Hamiltonian and all of whose probabilities converge to one.

My question is about the arguments at the end of the proof in Subsection 2.3. I have understood the beginning of the subsection where the existence of many paths of maximal length which contain certain fixed sections $M_1$ and $M_2$ of vertices is shown. But then the existence of a series of sets of new endpoints $K_p^isubset M_1$ of growing sizes is shown. I don’t even understand the construction of the first set $K_p^1$ which “can be seen easily” using the logical opposite of condition $D_3$. Which sets $S_1$ and $S_2$ is condition $neg D_3$ applied to? I suspect that $neg D_3$ is applied several times to different choices of $S_1$ and $S_2$ in order to get a disjoint union $bigcup_{pin L_1} K_p^1$. Or, are these sets automatically disjoint for some other reason? The choices of the sets $K^2_p$ and so on become even more mysterious to me. How is $neg D_4$ used, and which set $S$ is it applied to?

After that I am fine again: the same thing is done from the other end with $M_2$, and finally the cycle is obtained leading to a contradiction.

Thank you for reading that far. Some help or some hints would be appreciated very much.