We present a short exposition of the following results by S. Parsa. Let L be a graph such that the join L * {1, 2, 3} (i.e. the union of three cones over L along their common bases) piecewise linearly (PL) embeds into R 4 . Then L admits a PL embedding into R 3 such that any two disjoint cycles have zero linking number.There is C such that every 2-dimensional simplicial complex having n vertices and embeddable into R 4 contains less than Cn 5/3 simplices of dimension 2.We also present corrected statement and proof of the analogue of the second result for intrinsic linking.This paper provides short proofs of Theorems 1, 3 and 4 below. It can be considered as a complement to Zentralblatt review of [Pa15]. Denote [N] := {1, . . . , N}.Theorem 1 (S. Parsa). Let L be a graph such that the join L * [3] (i.e. the union of three cones over L along their common bases) piecewise linearly (PL) embeds into R 4 . Then L admits a PL embedding into R 3 such that any two disjoint cycles have zero linking number. This is formally a corollary of [Pa15, Theorem 1] but essentially a restatement of [Pa15, Theorem 1] accessible to non-specialists. Theorem 1 trivially generalizes to a d-dimensional finite simplicial complex L and embeddings L → R 2d+1 , L * [3] → R 2d+2 . See also Remark 7. Proof of Theorem 1. Consider L * [3] as a subset of R 4 . Take a small general position 4-dimensional PL ball ∆ 4 containing the point ∅ * 1 ∈ R 4 . Then the intersection ∂∆ 4 ∩ (L * [3]) is PL homeomorphic to L. Let us prove that this very embedding of L into the 3-dimensional sphere ∂∆ 4 satisfies the required property. Take any two disjoint oriented closed polygonal lines β, γ ⊂ ∂∆ 4 ∩ (L * [3]) ∼ = L. Then (β * {1, 2}) − Int ∆ 4 and (γ * {1, 3}) − Int ∆ 4 are two disjoint 2-dimensional PL disks in R 4 − ∆ 4 whose bound aries are β and γ. Hence β and γ have zero linking number in the 3-dimensional sphere ∂∆ 4 (by the following well-known lemma applied to the 4-dimensional ball S 4 − Int ∆ 4 ).Lemma 2. If two disjoint oriented closed polygonal lines in the 3-dimensional sphere ∂D 4 bound two disjoint 2-dimensional PL disks in the 4-dimensional ball D 4 , then the polygonal lines have zero integer linking number in ∂D 4 . I am grateful to A. Kupavskii, M. Skopenkov, M. Tancer and S. Parsa for useful private discussions, and to S. Parsa for public discussion presented in Remark 7.b.