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In my paper, I have used the expression "It is well known that" twice in a row. Using the same expression repeatedly may appear monotonous. Are there any other commonly used alternatives for this expression?

It is well known that every planar graph with $n\,(\ge3)$ vertices has at most $3n-6$ edges, and thus has the minimum degree at most 5. It is known that any planar graph with minimum degree 5 has at least 12 vertices and any planar graph of odd order with minimum degree 5 has at least 15.

I attempted to ask GPT, and its responses seem uncommon in mathematical research papers.

It suggests I use the following sentences:

"It is a known fact that..."

"It is widely recognized that..."

"It is an established result that..."

"It is commonly understood that..."

"It is widely accepted that..."

Note:

Yes, this question is better suited for English Stack Exchange. It should be deleted, but after seeing the nice suggestions provided by Will Jagy, I decided to keep it.

By the way, the suggestions of Xander Henderson and Greg Martin are very helpful; citing references is better than using "It is well known."

licheng
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    In the second sentence, replace "it is known" with "it is also known". That's an opinion. I am not an editor. – John Douma Sep 05 '23 at 02:43
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    This really doesn't seem like a question about mathematics. Also, I would avoid these kinds of phrases... They seem condescending to me. Why not simply state the fact and give a citation? For ensure "Every planar graph with $n\ge 3$ vertices had at most $3n-6$ edges [citation]". – Xander Henderson Sep 05 '23 at 02:44
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    Just because it is well known to some people does not make it well known to all. Just add a reference. – copper.hat Sep 05 '23 at 02:47
  • @XanderHenderson The theorem that the number of edges in a planar graph with $n\ge 3$ vertices is at most $3n-6$ is nearly standard, mentioned in (most) all graph theory textbooks, but I can hardly find it in the earliest literature. So, I'm not sure which reference would be more appropriate to cite, or if any textbook would suffice. – licheng Sep 05 '23 at 02:57
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    It's better to give an easy-to-find reference (a textbook for example) than an earlier but obscure reference. (If you know who proved it, say their name, but it's okay if you cite a more modern source.) As authors, we need to give the reader enough information so that they know where to go to verify the statement and learn more about it if they want. – Greg Martin Sep 05 '23 at 02:59
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    @XanderHenderson Yes, this question is better suited for English Stack Exchange. – licheng Sep 05 '23 at 03:59
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    @licheng If it is a standard result, it is fine to cite some textbook. Indeed, it might even be better---if I am reading a paper, and some result is given, I can get some idea about how deep, or "well-known", or obscure some result is based on the reference cited. In my own field, a reference to Folland's Real Analysis is going to indicate some very basic result; a reference to Hutchinson's 1981 paper is going to be a little more specialized but probably basic to the field, and a reference to Fraser's work is going to be pretty specialized. – Xander Henderson Sep 05 '23 at 12:34

1 Answers1

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It is well known that every planar graph with $n\,(\ge3)$ vertices has at most $3n-6$ edges, therefore having minimum degree at most 5. A planar graph with minimum degree 5 has at least 12 vertices. Finally, a planar graph of odd order, with minimum degree 5, has at least 15 vertices.

Will Jagy
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