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.