Disclaimer: I do not own Jareth, the Labyrinth, or anything else from the movie.
Something, made of vertices and edges of light, came flying up to them and hovered in the air before them. Magdalene looked at it. "Petersen's Graph!" she said.
"What?" Susan asked. "Lene, what's going on?"
She knelt and took Susan's hands. "Susan, this next part looks like graph theory to me. A graph is a set ofthings, any kind, where some are attached to each other or next to each other in some way. That out there," and she waved her hand at the void and the dancing lights, "is the easiest way to think about them, but mathematically, it's the same as, say, a family tree, or a map of the London Underground-you don't care about where the stations are, or how far away they are, you only care about whether a train connects them."
"I still don't understand," said Susan.
"Don't worry about it. I think I'll have to take care of this one." Lene stood up. "If I can figure out what to do," she added.
"Get to the door," said something.
Lene looked around. "Who said that?"
"Me. The Petersen Graph, or the starry thing hanging in front of you, if you don't like technical terms. Call me Star, it's shorter."
Lene turned to the Petersen Graph. It hovered in front of her: ten dots, five linked into a star, the other five linked into a pentagon. "How do I get to the door?"
"You see that vertex at your feet?" the graph asked. "All you have to do is connect it to the vertex in front of the door, and you and your friends can walk across to it."
"How do I do that? And for that matter, how do I connect things in here?"
"The same way you do anything in mathematics, of course. Proofs."
"Proofs about what?"
"In this case, Ramsey colorings. You have to start here, at Ramsey (2,2), and work your way across to Ramsey (4,4), in front of the door. There's vertices for all of the pairs of numbers, you know, and if you get the correct limit on the size of the Ramsey Graph, we'll make an edge to the next one."
"Right."
"What's a Ramsey coloring?" Dan asked.
"How large of a group of people do you need to ensure either m mutual friends or n mutual strangers?" Lene replied.
"What does that have to do with those?" he demanded, pointing to the graphs in the void.
"You have a vertex-point-for each person, and you draw an edge between each vertex. Color each edge red if they're strangers, blue if they're friends."
She paused. "If you've got enough dots, you'll have to get either a red Kn or a blue Km. A Kn is n vertices with an edge between any pair of them. What I need to do, it sounds like, is to figure out how many large enough' is, if you want to make sure there's a K4 of one color or a K4 of the other."
"Is that a technical explanation?" the Star asked.
"Almost. Be quiet, please. I want to do this right." She frowned. "WellI can get all of the (2, n)'s. It's just n."
"Prove it," the graph challenged.
"If it has any blue edges at all, we're done. Otherwise, it's all red, and you need exactly n vertices to force a red Kn."
Edges blinked into existence. "Good girl," the Star said approvingly. "You'd better take them in order, though. The next one is (3,3)."
"Well-um. I think-um, K6?"
"You tell me," the Star said uncompromisingly. "But move so you're standing on (2,3) first-it's the one attached to the vertex in front of the door. You'll need it, and you can only use the ones you're on."
Lene swallowed and stepped onto the first vertex. It seemed very small to be walking on-and she'd always been afraid of heights. She gritted her teeth and moved her other foot onto the globe of light, holding her hands out to keep her balance.
It wasn't necessary. The vertex seemed to have its own gravity. She slid around it, so that her brother and sisters were below her feet. She didn't feet inverted at all-the vertex was still down. It was still terrifying, being over that void, but she felt better knowing that she wouldn't fall off. She gritted her teeth and walked along the edge to the next vertex, the one the Star said was (2,3). She looked across the void-another vertex hung there, glowing. It had to be (3,3).
Lene turned back and beckoned for her brother and sisters to follow her. They did, Dan and Natalie gripping Susan's hands tightly. Lene turned back and concentrated on her proof.
"Well-each vertex has five edges coming from it-so pick a vertex, and at least three edges will be the same color. Call it blue. Look at the vertices on the ends of those edges-either they form a red K3, or there's a blue edge-in which case, it makes a blue K3 with the edges to the first vertex. So yes, I'll say six."
An edge formed between Lene and the vertex she'd seen floating in front of her. Without any prompting from the Star, she walked across it.
"Next, try (3,4)," the Star said.
