Randomized Construction of Complexes with Large Diameter

We consider the question of the largest possible combinatorial diameter among pure dimensional and strongly connected $$(d-1)$$ ( d - 1 ) -dimensional simplicial complexes on n vertices, denoted $$H_s(n, d)$$ H s ( n , d ) . Using a probabilistic construction we give a new lower bound on $$H_s(n, d)$$ H s ( n , d ) that is within an $$O(d^2)$$ O ( d 2 ) factor of the upper bound. This improves on the previously best known lower bound which was within a factor of $$e^{\varTheta (d)}$$ e Θ ( d ) of the upper bound. We also make a similar improvement in the case of pseudomanifolds.