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Looking through meta-MO I have found a proposal of Martin Sleziak to create a new "small-uncountable-cardinals", which I liked since many of my (and not only my) questions fall under this tag. So, I created a new tag "small-uncountable-cardinal" and tagged several my question with this new tag till realized that I forgot to add "s" at the end of cardinals. So, I tried to create a new (and more correct) tag "small-uncountable-cardinals", but the system has blocked this attempt writing that a similar tag "small-uncountable-cardinal" exists already. So, I removed all tags "small-uncountable-cardinal" from my questions and now no question is tagged with "small-uncountable-cardinal", but still this tag exists in the system and does not allow me to create a new tag "small-uncountable-cardinals". According to rules of MO, a tag which is not used for 24 hours will disappear. This means that I can hope to create the new tag "small-uncountable-cardinals" after 24 will pass. In the meantime I would like to ask the MO-community for opinion concerning the idea (basically of Martin Sleziak) to create this new tag "small-uncountable-cardinals" and retag the relevant questions (this last procedure is not accepted well by many MO users as it bumps up the retagged questions).

Search through MO by the phrase "small uncountable cardinal" yields 100 answers and I suggest that there are many questions which do not use this phrase but use small uncountable cardinals (like $\mathfrak b,\mathfrak p,\mathfrak d,\mathfrak c$, etc.). Do you this is a sufficient number for creating a tag or not?

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    $\begingroup$ "small uncountable cardinals" yields 100 answers when typed without quotation marks. This means questions in which each word occurs in the thread... The number of questions about small uncountable cardinals is smaller. $\endgroup$ – YCor Mar 12 at 11:46
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    $\begingroup$ @YCore To be precise, it shows up 4+4=8 results for "small uncountable cardinal" and "small uncountable cardinals", indeed many of those 100 results do not relate to small uncountable cardinals. $\endgroup$ – Taras Banakh Mar 12 at 11:49
  • $\begingroup$ @YCor To me a more reasonable estimate seems to be by checking the posts about specific cardinal, for example there are 22 questions in tag (set-theory) mentioning $\mathfrak p$, most of them use this symbol in the sense of cardinal $\mathfrak p$. (There are a few more links in the original post, perhaps I should have restricted them to questions.) $\endgroup$ – Martin Sleziak Mar 12 at 12:01
  • $\begingroup$ Since I see that you were criticized on the main site for bumping several old posts, I will add a link to this older discussion: Do we have an unofficial quota on how many old questions one should bump for minor edits in a single day? (Several of the questions linked there and questions tagged bumping are about similar problems.) Sorry for the digression - this is not directly related to the creation of the new tag. $\endgroup$ – Martin Sleziak Mar 12 at 12:16
  • $\begingroup$ @MartinSleziak I have check for $\mathfrak b$, $\mathfrak p$, $\mathfrak d$, "cardinal characteristic of the continuus", indeed, there more-or-less 100 post mentioning these cardinal charactistics in the proper sense. 874 poset mention $\mathfrak p$ but many of them in the different sense (line prime ideals etc) $\endgroup$ – Taras Banakh Mar 12 at 12:19
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    $\begingroup$ There is a separate question about the infinite-combinatorics. I would be OK with this being a broad tag that covers both Ramsey-type infinite combinatorics along with set-theoretic combinatorics and cardinal invariants. There's no tag wiki at this point - do other people have thoughts on what this tag should cover? $\endgroup$ – Carl Mummert Mar 12 at 15:13
  • $\begingroup$ Yes "infinite-combinatorics" is a broader field which (almost) swallows "small-uncountable-cardinals" and contains a lot of other things. For me "small-uncountable-cardinals" are synonyms to "cardinal characteristics of the continuum". $\endgroup$ – Taras Banakh Mar 12 at 16:30
  • $\begingroup$ "Infinite-combinatorics" would rather describe methods where "small-uncountable-cardinals" -- objects. $\endgroup$ – Taras Banakh Mar 12 at 16:37
  • $\begingroup$ @Taras Banakh - I would also include combinatorics on $\omega$, such as Hindman's theorem and Szemerédi's theorem, which have a different flavor from cardinal characteristics. $\endgroup$ – Carl Mummert Mar 12 at 17:52
  • $\begingroup$ @CarlMummert please could you link to that separate question about infinite-combinatorics? $\endgroup$ – YCor Mar 12 at 20:41
  • $\begingroup$ @YCor: I'm sorry - I just meant it is another issue related to the topic here $\endgroup$ – Carl Mummert Mar 12 at 21:14
  • $\begingroup$ @CarlMummert OK I see; however my belief is that it's not just a separate discussion, because the benefit of a tag depends on the closely related tags. $\endgroup$ – YCor Mar 12 at 21:17
  • $\begingroup$ Andrés Caicedo has removed small-uncountable-cardinals from mathoverflow.net/questions/40686 "variants of Martin's axiom at $\omega_1$". I would like to know participants' opinion about this. $\endgroup$ – YCor 2 days ago
  • $\begingroup$ @YCor Yes, I think that that post (about MA) do not fit very well to "small-uncountable-cardinals" because there is such cardinals in this question. $\endgroup$ – Taras Banakh 2 days ago
  • $\begingroup$ @TarasBanakh do you mean "because there is no such cardinals"? $\endgroup$ – YCor 2 days ago
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It seems that these three tags are being discussed in this thread: , , .The tag exists for a long time, but it might be good to improve the tag-info. Whether or not the other two tags should be created is part of this discussion - again, clarifying the content of the tag and suggesting the tag-info might be useful in deciding whether the tag might useful.

Let us use this community-wiki answer to write down some suggestions for the tag-info. (And in the comments we can discuss what should be included.) The post is made CW explicitly to encourage editing by other users. (After all, this is the purpose why the community-wiki feature exists.) So do not hesitate to edit the proposed tag-infos if you have any improvements or additions.

The tag (infinite-combinatorics)

This tag currently has a short tag-excerpt and empty tag-wiki.

Infinite combinatorics deals with various combinatorial properties of infinite sets. The topics might include, for example,

  • Ramsey theory on countably infinite sets, including results related to Szemerédi's theorem, Hindman's theorem, etc.
  • Ramsey theory on uncountable sets, such as the Erdős–Rado theorem, and partition calculus
  • Diamond ($\diamondsuit$) principles and relatives (such as $\clubsuit$), square ($\Box$) principles, club-guessing principles
  • Combinatorial properties of infinite graphs or partial orders (such as their chromatic number, marriage problems, etc)
  • Cardinal characteristic of the continuum and related topics
  • Infinite trees, such as Kurepa trees or Aronszajn trees
  • Ramsey ultrafilters, p-points and related topics
  • (Maximal) almost disjoint families

It may be desirable to rename this (infinitary-combinatorics). The tag exists in math.stackexchange with excerpt

For topics of a combinatorial character in set theory. Topics belonging to "combinatorial set theory" may be tagged this way. These include: Partition calculus, diamond principles, square principles, combinatorial properties of infinite graphs or partial orders, etc.

and tag info

This tag is for topics of a combinatorial character studied in set theory. Topics belonging to "combinatorial set theory" or "infinitary combinatorics" may be tagged this way. These include: Partition calculus (generalizations of Ramsey theory to infinite cardinals, infinite ordinals, other partially ordered structures, etc), diamond ($\diamondsuit$) principles and relatives (such as $\clubsuit$), square ($\Box$) principles, club-guessing principles, combinatorial properties of infinite graphs or partial orders (such as their chromatic number, marriage problems, etc), among others.

(The name infinite-combinatorics was briefly used over there when the tag was first being discussed, simply due to the limitation on the number of characters on tag names that we used to have. We switched to infinitary-combinatorics when we saw that limitation was no longer in effect.)

The tag (small-uncountable-cardinals)

The tag currently has a short tag excerpt and empty tag-info. (The tag was created very recently - it is possible that, depending on the outcome of this discussion, it might be either removed or possibly moderators might rename this tag.)

Small uncountable cardinals or cardinal characteristics of continuum are various cardinals which are typically between $\aleph_1$ and $2^{\aleph_0}$ and their definition often has a combinatorial flavor. Some examples are:

  • The cardinal $\mathfrak p$ - the smallest cardinality of subsystem of $[\omega]^\omega$ with strong finite intersection property and no pseudointersection.
  • Various cardinals related to $(\omega^\omega,\le^*)$ such as the bounding number $\mathfrak b$ (=the smallest cardinality of an unbounded subset) or the dominating number $\mathfrak d$ (=the smallest cardinality of dominating subset).

See also: Cardinal characteristic of the continuum on Wikipedia.

The tag (cardinal-characteristics)

A suggestion for the tag-info:

Cardinal characteristics of continuum are various cardinals which are typically between $\aleph_1$ and $2^{\aleph_0}$ and their definition often has a combinatorial flavor. Some examples are:

  • The cardinal $\mathfrak p$ - the smallest cardinality of subsystem of $[\omega]^\omega$ with strong finite intersection property and no pseudointersection.
  • Various cardinals related to $(\omega^\omega,\le^*)$ such as the bounding number $\mathfrak b$ (=the smallest cardinality of an unbounded subset) or the dominating number $\mathfrak d$ (=the smallest cardinality of dominating subset).

See also: Cardinal characteristic of the continuum on Wikipedia.

The tag also encompasses analogues at larger cardinals, such as the bounding number $\mathfrak b(\kappa)$ defined in terms of families of functions from $\kappa$ to $\kappa$, and cardinal invariants of certain structures (such as topological spaces or Boolean algebras).

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  • $\begingroup$ There are many people around here who are much more qualified to comment on the content of these tags than me. But I have tried to make at least some initial suggestions to get the process started. (If we want the community to create some description for these tags, it is probably easier to do on meta - where people can leave comments. Moreover, for the tags which don't exists yet, there is not tag-info which could be edited. $\endgroup$ – Martin Sleziak Mar 14 at 13:08
  • $\begingroup$ It is possible that as the outcome of this discussion there will be a separate tag for cardinal characteristics of continuum. A natural question is whether in such case - if such tag is indeed created - instead of mentioning this as a topic which should fall under (infinite-combinatorics) the tag-info should say that there is a separate tag for such questions. $\endgroup$ – Martin Sleziak Mar 14 at 14:20
  • $\begingroup$ I added some potential tag info for the last suggestion. $\endgroup$ – Andrés E. Caicedo Mar 14 at 19:15
  • $\begingroup$ Thanks for doing that @AndrésE.Caicedo. I hope this helps clarifying the indented scope of this tag. (It goes without saying that you should feel free to edit also the proposals for other tag-infos if you have some suggestions - these are definitely topics about which you know a lot more than I do.) $\endgroup$ – Martin Sleziak Mar 14 at 19:20
  • $\begingroup$ OK, I added some additional information regarding the first tag (the tag info from the sister site). $\endgroup$ – Andrés E. Caicedo Mar 14 at 19:28
  • $\begingroup$ What about two or three more closely related tags to add some entropy? $\endgroup$ – YCor Mar 14 at 22:15
  • $\begingroup$ @YCor Sorry, but I do not understand your comment. Nobody is proposing creation of several tags. The way I see it, this discussion might end up with some of the possibilities; A) No new tag is created, the tag-info for (infinite-combinatorics) is improved. B) A new tag called (small-uncountable-cardinals) is created. C) A new tag called (cardinal-characteristics) is created. (In either of the cases, at most one new tag is created.) $\endgroup$ – Martin Sleziak Mar 15 at 1:41
  • $\begingroup$ @AndrésE.Caicedo So if I understand it correctly, based on the suggested tag-info (current revision), the proposed (cardinal-characteristics) tag would be much wider than (small-uncountable-cardinals) and it would include various cardinal functions. So, for example, questions about tightness or weight of a topological space or questions about cellularity of a Boolean algebra would fit under this tag. Is that fair representation of the proposed (cardinal-charcteristics) tag? $\endgroup$ – Martin Sleziak Mar 15 at 2:18
  • $\begingroup$ I think so, sure. Or we could just call it cardinal-characteristics-of-the-continuum or cardinal-invariants-of-the-continuum if limiting the scope is important. In any case my point is that deciding that "small uncountable" means of size at most continuum does not make it so (and it is meaningless. Would the tag be applicable to a question about $\omega_4$, or only if we add the silly extra clause that the continuum is larger?), and saying that the scope of a "small-uncountable-cardinals" tag is precisely "cardinal-characteristics-of-the-continuum" is silly, better to use proper terminology. $\endgroup$ – Andrés E. Caicedo Mar 15 at 3:01
  • $\begingroup$ @AndrésE.Caicedo A minor problem with (cardinal-characteristics-of-the-continuum) is that it runs above 35 characters limit. In any case, thanks for clarifications on what the tag you're proposing encompasses. (Let's hope there will be some feedback from other users to see what they think about the proposals made in the question and the answers posted so far.) $\endgroup$ – Martin Sleziak Mar 15 at 3:05
  • $\begingroup$ @MartinSleziak indeed, my misunderstanding. Thanks for your effort to settle the question. $\endgroup$ – YCor Mar 15 at 8:02
  • $\begingroup$ I made a tweak to infinite-combinatorics; I think "Ramsey theory" has a different connotation in the countable case than the uncountable case, because the way the theorem is generalized is different (e.g. additive combinatorics vs. combinatorial set theory) $\endgroup$ – Carl Mummert Mar 16 at 13:40
  • $\begingroup$ Thanks for your edit @Carl Mummert. The purpose of this post is precisely that - to get some input on what should be in the tag-info and perhaps eventually - maybe after one or two weeks - when new feedback is being added - edit this into the actual tag-wiki for (infinite-combinatorics). (And also for some of the other tags - if the outcome of the discussion here is that they should be created.) $\endgroup$ – Martin Sleziak Mar 16 at 13:51
  • $\begingroup$ Hmm - I had gone ahead and copied the draft from here to there, since having it empty seemed suboptimal. Maybe that was too fast. I do think changes from here should be copied over again as this is updated. $\endgroup$ – Carl Mummert Mar 16 at 13:53
  • $\begingroup$ @Carl Mummert I don't think that there is any problem with that. As you say, if other users suggest further changes, the tag-wiki can be edited again. $\endgroup$ – Martin Sleziak Mar 16 at 13:57
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Unlike what another answer suggests, I think that "continuum hypothesis" is not a good tag to use as a proxy here.

Instead, I assume many of these questions would be about "cardinal characteristics", so maybe we need such a tag. It seems to me that "cardinal characteristics" would be more useful than the proposed "small uncountable cardinals", which seems to me to be a bit vague in scope. I also don't see what the advantage of using it be. For instance, many results involving forcing axioms are about the combinatorics at $\aleph_1$ and $\aleph_2$. I don't see any gain in having this tag added to questions about such results.

I understand that "cardinal characteristics" may be specific enough that a few questions that would use the "vague" tag may be left out, but it seems like a useful compromise.

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  • $\begingroup$ There are two things in small-uncountable-tags that would be unclear in a cardinal-characteristics: first, it would encompass large cardinals, and there's already the tag large-cardinals which is much bigger than the previous two ones (currently 498 questions). Second, it might be misused for counting problems finite combinatorics. $\endgroup$ – YCor Mar 12 at 20:04
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    $\begingroup$ I am confused by this comment. Why would one think that small uncountable cardinals is meant to encompass large cardinals? It is the same confusion I feel when reading your proposal. The tag CH is certainly not meant to encompass situations where CH fails. $\endgroup$ – Andrés E. Caicedo Mar 12 at 20:34
  • $\begingroup$ It seems I was confused by your answer, and thought you suggested cardinal-characteristics as better than small-uncountable-cardinals. If you think that creating small-uncountable-cardinals is a good idea, maybe it would have been a good starting point. $\endgroup$ – YCor Mar 12 at 20:39
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    $\begingroup$ @YCor No, I think that "small countable cardinals" is too vague in scope to be useful. Cardinal characteristics are specific objects and the examples suggested by Taras seem all to be about them. $\endgroup$ – Andrés E. Caicedo Mar 12 at 20:51
  • $\begingroup$ OK, so I maintain my previous comment: cardinal-characteristics would encompass the quite broad large-cardinals and this is at odds with the given proposal, if I understand correctly. $\endgroup$ – YCor Mar 12 at 20:55
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    $\begingroup$ @YCor No, that sounds crazy. "Cardinal characteristic" is a specific thing with specific meaning. "Large cardinals" means something else. It is like saying that a tag "Euclidean geometry" would encompass "Artificial intelligence". $\endgroup$ – Andrés E. Caicedo Mar 12 at 20:57
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    $\begingroup$ I'm thinking of "characteristic" in the vulgar sense, in the same way as "cardinal features". What is meant by cardinal characteristics? Anyway, if there's a precise sense and my "vulgar" interpretation is mistaken, I guess others will do the same mistake too. $\endgroup$ – YCor Mar 12 at 20:58
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    $\begingroup$ @YCor Not the people for which the tag is intended. Yes, we get questions on this site about "algebraic geometry" which are really about finding the equation of a line going through two points whose coordinates are given. This does not mean "algebraic geometry" is a useless tag. Classically, cardinal characteristics or cardinal invariants are cardinals $\kappa$ whose definition implies $\aleph_0<\kappa\le 2^{\aleph_0}$. In practice, they are defined in terms of combinatorial properties. $\endgroup$ – Andrés E. Caicedo Mar 12 at 21:08
  • $\begingroup$ Got it: en.wikipedia.org/wiki/Cardinal_characteristic_of_the_continuum. This will lead anyway, to numerous confusions. I also thought of algebraic geometry, but you probably know that algebraic geometry is by far a broader, more important and older subject, and a whole subfield of math. $\endgroup$ – YCor Mar 12 at 21:08
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    $\begingroup$ @YCor For instance, $\mathfrak b$, the unbounded number, is the least cardinality of an unbounded subset of $\mathbb N^{\mathbb N}$ under the relation $f\le^* g\Longleftrightarrow \exists n\,\forall m>n\, f(m)\le g(m)$. $\endgroup$ – Andrés E. Caicedo Mar 12 at 21:08
  • $\begingroup$ Yes, we get questions on this site about "algebraic geometry" which are really <--- like questions at the level of elementary undergrad proofs which are tagged [formal-proof]... $\endgroup$ – David Roberts Mar 13 at 5:35
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    $\begingroup$ I have to take your word for it that the experts on set theory prefer "cardinal characteristics" to "small uncountable cardinals" but I don't understand it. Why "characteristics"? What does this use of "characteristics" have to do with any other mathematical or non-mathematical use of the word? At least the meaning of "small uncountable cardinals" is related in an obvious way to the meanings of the three words. (Historically I guess it comes from Hechler's 1972 paper "A dozen small uncountable cardinals".) Of course it's not clear from the words "small uncountable cardinals" that "small" means $\endgroup$ – bof Mar 14 at 9:25
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    $\begingroup$ precisely $\le\mathfrak c$. On the other hand, from the words "cardinal characteristics" it's not clear to us people in the street that it's not about cardinal invariants of topological spaces, or Boolean algebras, or Abelian groups, or anything at all. $\endgroup$ – bof Mar 14 at 9:27
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Given the discussion, I tend to be gradually convinced that creating is a good idea, and a good counterpart to the existing quite broad , which currently has almost 500 occurrences. I initially expressed that all this could be embedded into , but several people have argued against this and I'm fine with those arguments. I'd like, anyway, that one additional benefit of the discussion would be to clarify the role/meaning of the tag ; I don't think it's a separate issue as the intersection is significant.

Also, I'm against ,

  • because this will result in a misunderstanding of the its meaning (will be misused at many occasions), as it will be widely understood as "properties of cardinals", and I don't think that properly understanding the meaning of a tag should be a privilege for those very specialists of the given subjects,

  • because "characteristics" seems to be used only by a proper subcommunity among the people dealing with such cardinals (I can substantiate this claim upon request), so a few questions naturally fitting with this tag will not be tagged so (or later by other people) — a typical such tag in another area seems to be .

  • because the restriction to be $\le 2^{\aleph_0}$ makes it too restrictive. Cardinals such as $(2^{\aleph_0})^+$ or $2^{2^{\aleph_0}}$ should be considered as small, as opposed to large cardinals. Small/large is not a completely defined boundary (roughly, large would be at least the smallest inaccessible) but I don't think it's not a problem, and it's even better than setting artificial boundary.

  • because I can't detect any sensible argument making it better than .

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  • $\begingroup$ Thank you for supporting the idea of introducing the tag "small-uncountable cardinals". Concerning the boundary between small and large cardinals is very precise: the existence of small cardinals follows from ZFC whereas the existence of large cardinals (starting with strong inaccessible) is a stronger assumption than the consistency of ZFC. More precisely this is a boundary between large cardinals and non-large. The boundary between small and non-small is $\mathfrak c$, the cardinality of continuum. $\endgroup$ – Taras Banakh Mar 13 at 11:19
  • $\begingroup$ So, up to my understanding of the realm of uncountable cardinals it can be divided into three categories: "small" (those in the interval $[\omega_1,\mathfrak c]$, "intermediate" (larger than $\mathfrak c$ but existing in ZFC, and finally "large" -- which do not necessarily exist in ZFC. $\endgroup$ – Taras Banakh Mar 13 at 11:21
  • $\begingroup$ @TarasBanakh I meant "strongly inaccessible" in my post (this is what I learnt as "inaccessible"). But the interval $[...,c]$ is a bad interval for defining a tag, since $\aleph_2$ or $2^{\aleph_1}$ would be small in some models and not small in others. Also I don't think the categories in $[c,q[$, for $q$ the smallest strongly inaccessible (if it exists) deserves to be considered separately. I don't think there a common accepted definition of "small" and I think it's better this way. $\endgroup$ – YCor Mar 13 at 11:22
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    $\begingroup$ "infinite-combinatorics" descibes rather methods and results involving combiknatorial properties of infinite sets. It is situated somewhere in the union of Set Theory and Combinatorics. Eventually this tag also can be created to point at the essense of a problem (which concerns combinatorial properties of infinite sets). $\endgroup$ – Taras Banakh Mar 13 at 11:26
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    $\begingroup$ @YCore Anyway "small" relates to cardinalities of objects appearing in Classical Matehmatics: not very far from the continuum. If you try to label each natural mathematical object with its complexity in the Comulative Hierarchy of the Universum, you will discover that the most difficult objects of Functional Analysis and Differential Equations (like functions spaces of spaces of operators) have complexity not more $\omega+13$ (maybe $\omega+15$). Which means that the cardinality of such natural objects is at most $\beth_{15}$. This can be informally considered as the realm of small cardinals. $\endgroup$ – Taras Banakh Mar 13 at 11:31
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    $\begingroup$ @TarasBanakh Great, I've created a short usage guidance for infinite-combinatorics, which will hopefully be improved by better acquainted users. $\endgroup$ – YCor Mar 13 at 11:33
  • $\begingroup$ Very good. It is also correspond to my understanding what it is. $\endgroup$ – Taras Banakh Mar 13 at 11:34
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    $\begingroup$ @TarasBanakh great, I agree with you're last comment (on "small") — btw sorry readers for the double superposed discussion :) $\endgroup$ – YCor Mar 13 at 11:34
  • $\begingroup$ @TarasBanakh I created a usage guidance to the new tag, it's great if you have a look. $\endgroup$ – YCor Mar 13 at 22:42

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