For graphs F and G, let Fβ(G,G) signify that any red/blue edge
coloring of F contains a monochromatic G. Define Folkman number f(G;p) to
be the smallest order of a graph F such that Fβ(G,G) and Ο(F)β€p. It is shown that f(G;p)β€cn for graphs G of order n with
Ξ(G)β€Ξ, where Ξβ₯3, c=c(Ξ) and p=p(Ξ) are
positive constants.Comment: 11 page
For a given monic integral polynomial f(x) of degree n, we define local
roots riβ of f(x) for a completely decomposable prime p by riββZ, f(riβ)β‘0modp and 0β€r1ββ€r2ββ€β―β€rnβ<p. With numerical data, we propose a conjecture on the distribution of
(r1β/p,β¦,rnβ/p), which is a new kind of equi-distribution, and a
conjecture of the distribution of (r1β,β¦,rnβ) which satisfies riββ‘RiβmodL for given natural numbers L,R1β,β¦,Rnβ, which is nothing but
Dirichlet's theorem on an arithmetic progression in the case n=1
A function f:Β {β1,1}nβR is called pseudo-Boolean.
It is well-known that each pseudo-Boolean function f can be written as
f(x)=βIβFβf^β(I)ΟIβ(x), where ${\cal F}\subseteq \{I:\
I\subseteq [n]\},[n]=\{1,2,...,n\},and\chi_I(x)=\prod_{i\in I}x_iand\hat{f}(I)arenonβzeroreals.Thedegreeoffis\max \{|I|:\ I\in {\cal
F}\}andthewidthoffistheminimuminteger\rhosuchthateveryi\in
[n]appearsinatmost\rhosetsin\cal F.Fori\in [n],let\mathbf{x}_ibearandomvariabletakingvalues1orβ1uniformlyandindependentlyfromallothervariables\mathbf{x}_j,j\neq i.Let\mathbf{x}=(\mathbf{x}_1,...,\mathbf{x}_n).Thepβnormoffis||f||_p=(\mathbb E[|f(\mathbf{x})|^p])^{1/p}foranyp\ge 1.Itiswellβknownthat||f||_q\ge ||f||_pwheneverq> p\ge 1.However,thehighernormcanbeboundedbythelowernormtimesacoefficientnotdirectlydependingonf:iffisofdegreedandq> p>1then ||f||_q\le
(\frac{q-1}{p-1})^{d/2}||f||_p.ThisinequalityiscalledtheHypercontractiveInequality.Weshowthatonecanreplacedby\rhointheHypercontractiveInequalityforeachq> p\ge 2asfollows: ||f||_q\le
((2r)!\rho^{r-1})^{1/(2r)}||f||_p,wherer=\lceil q/2\rceil.Forthecaseq=4andp=2,whichisimportantinmanyapplications,weproveastrongerinequality: ||f||_4\le (2\rho+1)^{1/4}||f||_2.