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Beautiful article, though not sure why two examples are introduced (students x instruments, and jobs x hobbies). The abstract for Shitov's paper is also a gem
by FabHK 7y ago
Beautiful article, though not sure why two examples are introduced (students x instruments, and jobs x hobbies).
The abstract for Shitov's paper is also a gem of brevity, here reproduced in its entirety:
"The chromatic number of G×H can be smaller than the minimum of the chromatic numbers of finite simple graphs G and H."
(And, of course, the first reference in the paper is co-authored by Erdős.)
- pmiller2 7y agoI’ll take 2 excellent examples over a definition and one opaque and unmotivated example any day (which is what you get in a lot of papers).
- edjrage 7y agoThe text and the examples are good enough, but I'm sure just one example accompanied by some nice illustrations would be much more immediately illuminating.
- klyrs 7y agoBecause it's an article for non-graph-theorists. People tend to follow sentences involving familiar categories, than "nodes of H" and "nodes of G". And speaking as a graph theorist. When you've got three graphs like this, and you want to unambiguously refer to nodes of each, leaving things perfectly general can get quite tedious and torture the language. For example, sometimes it's just nicer to just say "red nodes" and "blue nodes" where any colors would suffice.
- FabHK 7y agoI should clarify that I didn't wonder why an example was introduced (which makes perfect sense), but why more than one was introduced (not sure that makes sense for this type of overview article).
- klyrs 7y agoMaybe to subtly discourage a misconception, that it's only about students x instruments? Kinda reaching here and I see what you mean
- zten 7y agoI don't think it's that big of a reach! I agree that it might be a bit unnecessary in this context, but in math classes it was always enormously valuable to have more examples than fewer ones. The pattern would become easier to identify. This helped out quite a bit if an example mixed in some other concept that I was quite a bit weaker at, because additional fully-worked examples would separate out all of the components and help me figure out what was blocking my full understanding of the problem.
- gpvos 7y agoWithout the examples, I wouldn't have understood the article. And I did have some graph theory way back in university.
- addicted 7y agoThe first example is used to explain what a tensor product of 2 graphs is. The second example was used to show how colouring the tensor product may actually be useful. I thought the second example was a little more complex (primarily because it requires the connection of incompatible nodes, as opposed to the more straightforward pairing of compatible nodes), so having a different, more straight forward example to just explain tensor products was helpful. Also, having 2 different unrelated examples for the same thing can definitely help with understanding the underlying property/theorem.