We will present problems (and solve some of them) about triangle-free graphs related to Erdős' Sparse Half Conjecture: Every triangle-free graph on n vertices has an induced subgraph on n/2 vertices with at most n^2/50 edges.
Among others we prove the following variant of it: For every sufficiently large even integer n the following holds. Every triangle-free graph on n vertices has a partition V(G)=A U B with |A|=|B|=n/2 such that e(G[A])+e(G[B]) <= n^2/16.