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Let $K\subset\mathbb{C}$ be a compact set in the plane whose logarithmiccapacity $c(K)$ is strictly positive. Let $\mathscr{P}_n(K)$ be the space ofmonic polynomials of degree $n,$ \emph{all} of whose zeros lie in $K.$ For$p\in \mathscr{P}_n(K),$ its filled \emph{unit leminscate} is defined by$\Lambda_p = \{z: |p(z)| < 1\}.$ Let $\mathcal{C}(\Lambda_p) $ denote thenumber of connected components of the open set $\Lambda_p,$ and define$\mathscr{C}_n(K) = \max_{p\in \mathscr{P}_n(K)}\mathcal{C}(\Lambda_p).$ Inthis paper we show that the quantity \[M(K) =\limsup_{n\to\infty}\dfrac{\mathscr{C}_n(K)}{n},\] satisfies $M(K) < 1$ whenthe logarithmic capacity $c(K) < 1,$ and $M(K) = 1$ when $c(K)\geq 1.$ Inparticular, this answers a question of Erd\"os et. al. posed in $1958$. Inaddition, we show that for nice enough compact sets whose capacity is strictlybigger than $\frac{1}{2}$, the quantity $m(K) =\liminf_{n\to\infty}\dfrac{\mathscr{C}_n(K)}{n} > 0.$

Number of components of polynomial lemniscates: a problem of Erd\"os, Herzog, and Piranian