I think your quote from Gallian doesn't address syzarian's example, actually. The two properties you quoted are about the fact the distributivity works whether you're multiplying on the left or on the right. That's one…
Note that for this property, you want edges to be reasonably close, but you also need to say that vertices are reasonably far apart: otherwise you could always get a "less tangled" graph by just shrinking it until it's…
Yes, if "untangled" means "no edge crossings", then 3 dimensions is enough for any graph to be untangled. As a proof, you can put the vertices at coordinates (0, 0, 0), (1, 1, 1), (2, 4, 8), (3, 9, 27), ..., (n, n^2,…
Unfortunately, constructive vs classical (vs linear, etc.) applies to proofs, but this is really about definitions. Proofs can be correct or incorrect pretty straightforwardly, but definitions being correct or not is…
Now you're reminding me of a wacky math conversation I had at Mathcamp [1] with a much smarter guy, who was talking about more esoteric definitions of volume in euclidean space. Something like: - n-dimensional volume is…
Yes, those two facts about zero/empty cases (and so many more) are definitely related, and this class of facts is one of my favourites! Usually, if you're dealing with something algebraic in flavour (which is a very…
I think your quote from Gallian doesn't address syzarian's example, actually. The two properties you quoted are about the fact the distributivity works whether you're multiplying on the left or on the right. That's one…
Note that for this property, you want edges to be reasonably close, but you also need to say that vertices are reasonably far apart: otherwise you could always get a "less tangled" graph by just shrinking it until it's…
Yes, if "untangled" means "no edge crossings", then 3 dimensions is enough for any graph to be untangled. As a proof, you can put the vertices at coordinates (0, 0, 0), (1, 1, 1), (2, 4, 8), (3, 9, 27), ..., (n, n^2,…
Unfortunately, constructive vs classical (vs linear, etc.) applies to proofs, but this is really about definitions. Proofs can be correct or incorrect pretty straightforwardly, but definitions being correct or not is…
Now you're reminding me of a wacky math conversation I had at Mathcamp [1] with a much smarter guy, who was talking about more esoteric definitions of volume in euclidean space. Something like: - n-dimensional volume is…
Yes, those two facts about zero/empty cases (and so many more) are definitely related, and this class of facts is one of my favourites! Usually, if you're dealing with something algebraic in flavour (which is a very…