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Here's how to make the last part of gowers's proof precise (the idea is from Gábor Simonyi). You have a complete graph of $M$ vertices covered by $t$ bipartite graphs such that each vertex is in at …

I found out that this is a known problem, and was solved in 1973. The Lovász: Combinatorical problems and exercises actually gives a solution in exercise 13.27. This gives asymptotically better esti …

Erich Friedman's packing center claims that you can't cover with 6 disks, and that this was proved by Károly Bezdek in 1979. If you want a more exact reference, ask Erich Friedman in email.

This event means that in the sequence of outcomes $ A_1, ..., A_r $ you don't have $ b $ adjacent falses. Suppose $ b \le t $. Let $ r-t $ be the index of the last true event in that sequence. Then …

A binary tree of which every subtree is search-optimal is called an AVL tree (height-balanced tree). Efficient algorithms for AVL trees are described in Knuth's The Art of Computer Programming chapte …

The Erdős-Ko-Rado theorem talks about how large an intersecting set system (a set of pairwise intersecting sets) can be if the size of the base set is fixed. I'm interested about intersecting set sys …

While you've already got a good answer, let me offer a worse one. Take a random deal of the cards (12 to each player), and let X be the number of card faces whose two cards go to different players. …

In the answer to Open problems in Euclidean geometry? , Alexey Ustinov brings into attention to a 2012 article. Greg Aloupis, Robert A. Hearn, Hirokazu Iwasawa, Ryuhei Uehara, Covering Points wi …

You say you are interested in small $ k $. This makes sense, because allowing an arbitrarily large $ k $ makes the question trivial (provided you allow repetition of a linear order with any multiplic …

I believe I can prove this with a standard Ramsey-type argument, though f will grow slower than linear. You'll need the following useful lemma. Lemma 1 (bipartite Ramsey). For any natural number …

Maxwell's equations were originally formulated for Newtonian physics. However, special relativity has found that these equations have a surprising symmetry to Lorentz transformations. The equations …

Wilfried Imrich, Sandi Klavžar, Product Graphs, Structure and Recognition gives this as theorem 8.38 on page 268 as well. I should have looked in this book earlier, but now Dan Stahlke's answer made …

I believe the book Hajnal Péter: Gráfelmélet. 1997, Polygon, Szeged. is an extended answer to exactly this question. (There's a second edition from 2003, but apparently no translations to oth …

No, and you can see this from just a counting argument. For determining which of the $ n $ chords of the circle intersect, it is enough to know the order of the $ 2n $ endpoints on the circle. (You …

I'm searching for a reference to a particular alternate definition of the fractional chromatic number of graphs. Let me review the most common definition and basic properties first. Let $ G $ be …