About a result by P. Erdős and H. Sachs on graph with large girth

Question

I recently came across the following result:

For any integers $d,r\geq 2$ and $n\geq 4d^{r(d+1)}$, there is a $d$-regular graph on $n$ vertices with girth at least $(d+1)r+1$.

referring to

"P. Erdős and H. Sachs, Reguläre Graphen gegebener Taillenweite mit minimaler Knotenzahl, Wiss. Z. MartinLuther-Univ. Halle-Wittenberg Math.-Natur. Reihe 12 (1963), 251–257."

This paper can be downloaded from https://users.renyi.hu/~p_erdos/1963-16.pdf.

Unfortunately, this was written in German. Would anyone be kind enough to explain it briefly in English? Besides, there are fancy results in the last century written in Russian, Germany, and France, etc. For those that are not translated in English, is there a useful way to read them?

I indeed tried Google translation but turns out to be like the following picture:

Ugh....

Answer 1

I OCR'd the text from the abstract (the pdf contains poor quality text, OCR'ing it yourself gives better results), pasted the OCR'ed text into DeepL, and received the following, without any edits from my side. This seems quite usable to me, if you read it together with the original that contains the formulas.

Regular graphs of given waist width with minimum number of nodes
PAUL ERDÖs and HORST SACHS*)
An estimation for the minimal number of nodes of regular graphs, which do not contain a circle of length < I Let G') be a graph with n vertices; vertices of G) are denoted by x, ..., y,..., edges are denoted by (2,, x,). Loops and bipartitions are not allowed. The valence (or order) 2 () of a vertex x of () is the number of edges that coincide with x. Let G(7) be a graph with the nodes X,, ..., Xn, which contains edge (x,, x,) but no edge (∞,, Xg); (G*") - (x,, x,) + (x,, x.)) is obtained from Eq) by omitting the edge (x,, x,) and adding the edge (2,, xg). A circle is a simply closed set of edges. Let the distance (or distance) e (G'*); x;, x,) from x, and x, in G'n) be the length of the shortest edge move connecting x; and x; (the length of an edge move or circle is the number of edges occurring in the edge train or circle). The length of the shortest circle in G we call the waist width of G'". Let K (x, r) be the set of all node- points which can be reached from x by an edge of length S r (we should actually write K (G'); x, r) but we do not do this, because we will use this symbol in a way that no misunderstanding can arise in this case). can arise). Let S be the number of elements of a set S.

As suggested by GH from MO I also tried chatGPT. Here is the output, again with no edits from my side.

Regular graphs of given waist width with minimal number of vertices
PAUL ERDÖS and HORST SACHS*)
An estimation for the minimal number of vertices of regular graphs, which do not contain a cycle of length < I (1). Let G'* be a graph with n vertices; vertices of G* are denoted by x, ..., y,..., edges are denoted by (x, y). Loops and bipartitions are not allowed. The valence (or order) of a vertex x in G* is the number of edges that coincide with x. Let G'(7) be a graph with the nodes X1, ..., Xn, which contains edge (x1, x2) but no edge (x1, Xg); (G'* - (x1, x2) + (x1, Xg)) is obtained from G* by omitting the edge (x1, x2) and adding the edge (x1, Xg). A circle is a simply closed set of edges. Let the distance (or distance) e (G') between x1 and x2 in G' be the length of the shortest edge move connecting x1 and x2 (the length of an edge move or circle is the number of edges occurring in the edge train or circle). The length of the shortest circle in G is called the waist width of G'. Let K(x, r) be the set of all node-points which can be reached from x by an edge of length ≤ r (we should actually write K(G; x, r) but we do not do this, because we will use this symbol in a way that no misunderstanding can arise in this case). Let S be the number of elements of a set S.

And to complete the comparison, here is Google Translate's output:

Regular graphs of given waist size with a minimum number of nodes
PAUL ERDÖs and HORST SACHS*)
An estimate for the minimum number of nodes of regular graphs, which do not contain a circle of length Let G') be a graph with n vertices; vertices of G) are denoted by x, ..., y,..., edges are denoted by (2,, x,). Loops and bipartitions are not allowed. The valence (or order) 2 () of a vertex x of () is the number of edges that coincide with x. Let G(7) be a graph with the nodes X,, ..., Xn, which contains edge (x,, x,) but no edge (∞,, Xg); (G*") - (x,, x,) + (x,, x.)) is obtained from Eq) by omitting the edge (x,, x,) and adding the edge (2,, xg). A circle is a simply closed set of edges. Let the distance (or distance) e (G'*); x;, x,) from x, and x, in G'n) be the length of the shortest edge move connecting x; and x; (the length of an edge move or circle is the number of edges occurring in the edge train or circle). The length of the shortest circle in G we call the waist width of G'". Let K (x, r) be the set of all nodes points which can be reached from x by an edge of length S r (we should actually write K (G'); x, r) but we do not do this, because we will use this symbol in a way that no misunderstanding can arise in this case). can arise). Let S be the number of elements of a set S.

I would say all three do a good job. DeepL has the feature that you can feed it an entire document in Word, and it will translate it while retaining the formatting. But that might not be necessary here.