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Logarithm $\log p$ of this product is also some definite integral (not a surprise, any number is a definite integral of appropriate function), but the integrand is more sophisticated than $x^{-x}$. …

There is no such $c$. Assume that you have a set with $n$ elements, and maximal $B$ has $k$ elements. Take two copies $C$, $D$ of $A$ with disjoint supports. Add initial segment $1,1,\dots,1$ of lengt …

For higher dimension, this does not hold, see the answer by Sergei Ivanov here. For dimension 2, in the closed set $K$ without lattice-point free translate (in other words, such that $K$ has a point i …

Yes, as Christian Remling says, the maximal size of $S$ is indeed $\sum_{i<d} \binom{n}i$. This is known as the shattering lemma (found independently by Sauer, Shelah-Pearls, and Vapnik-Chervonenkis), …

it is not what you are asking for (not a reference), but just maybe slightly easier proof, without searching for $K_{3,3}$. We may take any vertex $x$ of degree $d$, if any of $d$ vertices adjacent to …

Bolibruch's nice explanation is here: http://www.mccme.ru/free-books/dubna/bol1.pdf I do not know whether it has English translation, sorry.

This is the answer only to my own question in the comments. Two polynomials with real roots are not enough in general. For seeing this, assume that $f(z)=\lambda g(z)+\mu h(z)$ for complex numbers $ …

Let me explain how this may be solved without tricks, this may help for other similar issues. 1) The inequality must hold true for $\sum x_i\leqslant 1$, $\sum y_i\leqslant 1$ (instead of equalities …

This is rather a comment to Nate Eldredge's answer, which reduces the (negative) answer to the existence of a set $\Omega\subset \mathbb{R}^2$ which is not a countable intersection of countable unions …

This is not a reference, sorry. The algorithm looks quite straightforward. We require additionally that for the vertex sets $V_0=V(C),V_i=V(C\cup P_1\ldots\cup P_i), i=1,2,\ldots$ the restriction $T( …

Well, $f''f^2 +xf' ^3+2ff'^2 =0$ modulo 3 for $f=\prod(1-x^m)=\sum_{n\in \mathbb{Z} } (-1)^nx^{n(3n+1)/2}$ may be quickly seen as follows. Differentiating the power series for $f$ and expanding the b …

I think what you ask about is a partial case of Bateman–Horn conjecture for polynomials $f_1(x)=x,f_2(x)=4x+1$. The expected value of $C$ equals $$(1-1/2)(1-1/2)^{-2}\prod_{p>2}\left\{(1-2/p)(1-1/p)^{ …

There is an exercise in Stanley's Enumerative Combinatorics (Ex. 12 in Chapter 3 of Vol. 1): "True or false: if every chain and every antichain of a poset $P$ is finite, then $P$ is finite." It also c …

This is exactly Rado theorem on independent transversal (for the corresponding linear matroid). Strictly speaking, to use it you should deal with finite sets, but a dimension of a vector subspace alwa …

Denote these subsets which sum up to $s$ by $\alpha, \beta, \gamma$. Partition each subset $\alpha, \beta, \gamma$ onto two disjoint parts: $\alpha=\alpha_1\sqcup \alpha_2$,$\beta=\beta_1\sqcup \beta_ …