Mathematical Research in High School: The PRIMES Experience

Consider a finite set of lines in 3-space. A joint is a point where three of these lines (not lying in the same plane) intersect. If there are L lines, what is the largest possible number of joints? Well, let’s try our luck and randomly choose k planes. Any pair of planes produces a line, and any triple of planes, a joint. Thus, they produce L := k(k − 1)/2 lines and and J := k(k − 1)(k − 2)/6 joints. If k is large, J is about [[√2]/3]L[superscript 3/2]. For many years it was conjectured that one cannot do much better than that, in the sense that if L is large, then J ≤ CL[superscript 3/2], where C is a constant (clearly, C ≥ [√2]/3]). This was proved by Larry Guth and Nets Katz in 2007 and was a breakthrough in incidence geometry. Guth also showed that one can take C = 10. Can you do better? Yes! The best known result is that any number C > 4/3 will do. This was proved in 2014 by Joseph Zurer, an eleventh-grader from Rhode Island [Z]