Theorem 27 from our article [1] states that any closed triangulated surface S is balanced, i.e. $$\mu (X) \ge \mu (S)$$ for any subcomplex $$X\subset S$$. Here the notation $$\mu (Y)$$ stands for the ratio v / f where v and f are the numbers of vertices and faces (i.e. 2-simplexes) in a 2-complex Y correspondingly.

We noticed that the arguments of the proof of Theorem 27 are valid only under an additional assumption that $$\chi (S)\ge 0$$. The assumption $$\chi (S)\ge 0$$ is implicitly used in the sentence “Since $$f\ge f'$$ the above inequality follows from $$2-2b_1(S') +e_0\ge 4-4g$$ on page 134. Thus, Theorem 27 from [1] should read:

Any closed connected triangulated surface S with $$\chi (S) \ge 0$$ is balanced.

On the contrary, any closed triangulated surface S with negative Euler characteristic, $$\chi (S) <0$$ , admits a subdivision which is unbalanced. Indeed, let S be a triangulated closed surface with $$\chi (S) <0$$ and let X be obtained from S by removing the interior of a single 2-simplex $$\sigma$$. Then

\begin{aligned} v(X) =v(S) = f(S)/2 + \chi (S), \quad f(X)=f(S)-1 \end{aligned}

implying that

\begin{aligned} \mu (S)< \mu (X)<1/2, \end{aligned}
(1)

since $$\chi (S) < -1/2$$. Let $$S'$$ be obtained from S by subdividing the simplex $$\sigma$$ and introducing k new interior vertices in the interior of $$\sigma$$. Then

\begin{aligned} v(S') = v(S) +k, \quad f(S') = f(X) +2k +1 = f(S) +2k. \end{aligned}

Therefore,

\begin{aligned} \mu (S') = \frac{v(S)+k}{f(S)+2k} \end{aligned}

is approaching 1 / 2 for $$k\rightarrow \infty$$. Thus we may find k large enough so that $$\mu (S')>\mu (X)$$, in view of (1). This shows that $$S'$$ is unbalanced since X is a subcomplex of $$S'$$.

This correction affects only statement 5 of Corollary 28 from [1] which should be reformulated as follows:

If $$p\gg n^{-1/2 +\varepsilon }$$ for some $$\varepsilon >0$$ then, given a topological type of a closed surface, there exists $$f_0=f_0(\varepsilon )$$ such that any balanced triangulation of the surface having more than $$f_0$$ 2-simplexes will be simplicially embeddable into a random 2-complex Y , a.a.s. In particular, if $$p\gg n^{-1/2 +\varepsilon }$$ , a random 2-complex Y contains small closed orientable and nonorientable surfaces of all topological types, a.a.s.