A Two-Graph Refinement of Paulsen's Lollipop Bounds
Let $a_L(n)$ be the maximum number of regions into which $n$ lollipops divide the plane. Paulsen introduced a second obstruction for this problem, based on pairs of circles meeting at obtuse angle, in addition to the stem-direction obstruction of Cutler-Karlsson-Sloane. We recast Paulsen's argument as a weighted problem for two graphs: a $K_4$-free graph $D$ of non-close stem pairs and a $K_5$-free graph $E$ of non-intriguing circle pairs. For the total number $C$ of pairwise crossings, $$ C\le 4\binom n2+|D|+|E|+|D\cap E|. $$ Paulsen bounds the final term by $|D|$. We keep the overlap term and analyze near-extremal configurations of $D$ and $E$. This closes all of Paulsen's remaining gaps up to $n=17$, and also closes $n=19$: $$ \begin{array}{c} a_L(0),a_L(1),\ldots,a_L(17)\\ =1,2,10,25,45,71,104,142,186,237,294,356,425,500,580,667,761,859, \end{array} $$ and $$ a_L(19)=1076. $$ The same method gives the one-region gaps $$ 964\le a_L(18)\le965,\qquad 1193\le a_L(20)\le1194. $$