Large Simple d-Cycles in Simplicial Complexes

We show that the size of the largest simple d-cycle in a simplicial d-complex $K$ is at least a square root of $K$’s density. This generalizes a well-known classical result of Erd\H{o}s and Gallai \cite{EG59} for graphs. We use methods from matroid theory applied to combinatorial simplicial complexes.