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Series: Combinatorics Seminar

Series: Combinatorics Seminar

Series: Combinatorics Seminar

Series: Combinatorics Seminar

Series: Combinatorics Seminar

Series: Combinatorics Seminar

Series: Combinatorics Seminar

Series: Combinatorics Seminar

Series: Combinatorics Seminar

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.

Series: Combinatorics Seminar

For integers k>2 and \ell0, there exist \epsilon>0 and C>0 such that for sufficiently large n that is divisible by k-\ell,
the union of a k-uniform hypergraph with minimum vertex degree \alpha n^{k-1} and a binomial random k-uniform hypergraph G^{k}(n,p) on the same n-vertex set
with p\ge n^{-(k-\ell)-\epsilon} for \ell\ge 2 and p\ge C n^{-(k-1)} for \ell=1 contains a Hamiltonian \ell-cycle with high probability. Our result is best possible up to
the values of \epsilon and C and completely answers a question of Krivelevich, Kwan and Sudakov.
This is a joint work with Jie Han.