Abstract: In 1973 Erdos asked whether there are n-vertex partial Steiner triple systems with arbitrary high girth and quadratically many triples. (Here girth is defined as the smallest integer g ge 4 for which some g-element vertex-set contains at least g-2 triples.)
We answer this question, by showing existence of approximate Steiner triple systems with arbitrary high girth. More concretely, for any fixed ell ge 4 we show that a natural constrained random process typically produces a partial Steiner triple system with (1/6-o(1))n^2 triples and girth larger than ell. The process iteratively adds random triples subject to the constraint that the girth remains larger than ell. Our result is best possible up to the o(1)-term, which is a negative power of n.
Joint work with Tom Bohman.
Biography: Lutz Warnke is an assistant professor in Mathematics at Georgia Tech; he received his PhD from University of Oxford. His research area is probabilistic combinatorics and random discrete structures, with particular interest in random graph theory and phase transitions.
For his work he received in 2014 the Richard-Rado-Prize (German Mathematical Society), in 2016 the Denes Konig Prize (SIAM), and in 2018 a Sloan Research Fellowship.