In 2019 P. Patak and M. Tancer obtained the following higher-dimensional generalization of the Heawood inequality on embeddings of graphs into surfaces. We present a short well-structured proof accessible to non-specialists in the field. Let $Δ_n^k$ be the union of $k$-dimensional faces of the $n$-dimensional simplex. Theorem. (a) If $Δ_n^k$ PL embeds into the connected sum of $g$ copies of the Cartesian product $S^k\times S^k$ of two $k$-dimensional spheres, then $g\ge\dfrac{n-2k-1}{k+2}$. (b) If $Δ_n^k$ PL embeds into a closed $(k-1)$-connected PL $2k$-manifold $M$, then $(-1)^k(χ(M)-2)\ge\dfrac{n-2k-1}{k+1}$.

翻译：2019年，P. Patak与M. Tancer获得了关于图嵌入曲面问题的Heawood不等式的高维推广。本文提出一个结构清晰、简明易懂的证明，使该领域非专业人士也能理解。令$Δ_n^k$表示$n$维单形中所有$k$维面的并集。定理：(a) 若$Δ_n^k$可分段线性嵌入到$g$个$k$维球面笛卡尔积$S^k\times S^k$的连通和中，则$g\ge\dfrac{n-2k-1}{k+2}$。(b) 若$Δ_n^k$可分段线性嵌入到闭的$(k-1)$连通分段线性$2k$维流形$M$中，则$(-1)^k(χ(M)-2)\ge\dfrac{n-2k-1}{k+1}$。