Among many problems Erdos posed was the one about the maximal possible length of the lemniscate $|p(z)|=1$ where $p$ is a monic polynomial of degree $n$. His conjecture was that the worst polynomial is $z^n-1$ for which the length in question equals $2n+O(1)$. The best upper bound up to date was due to Eremenko and Hayman (slightly greater than $9n$). We will prove an estimate that is much closer to the conjectured $2n+O(1)$ for large $n$ than the Eremenko-Hayman one. I intend to present a full proof (except, maybe, a couple of small technical details). This is a joint work with Alexandre Fryntov.