Monochromatic Solution And Density Of Equation On The Integers

Monochromatic solution and density of equation on the integers have been studied widely in combinatorial number theory.A k-partition ?A1,A2,...,Ak?of the set[N]= {1,2,.,.,N} is also called a k-coloring of[N]Given a polyno-mial f(z)of integer coefficients and a k-coloring of[N],we say that there is a monochromatic solution of the equation a1x1 + a2x2 +…+amxm = f(z)if there exist pairwise distinct monochromatic x1,x2,...,xm ?[N]such that the above equation holds for some z ? Z.Problems of this type are often referred to as Ramsey-type problems.Ramsey-type problems on the integers derive from the celebrated Schur's Theorem:For all k-coloring of positive integers,the equation x + y-z = 0 has the monochromatic solution.In 2006,Khalfalah and Szemeredi proved the conjecture of Roth,Erdos,Sarkdzy and Sos:Suppose f(z)is a non-constant polynomial of integer coefficients with positive leading coefficient and 2|f(z)for some integer z.If N is sufficiently large,then for each k-coloring of[N],there exists a distinct monochromatic pair {x,y} such that x + y = f(z)for some integer z.In Chapter 2 of this thesis,we generalize the result of Khalfalah and Szemeredi and prove the following theorem.Theorem 1.Given integers k?2,n?2,m?2 and a1,a2)...,am ?Z\{0}.Let f(z)=(?)be a polynomial of integer coefficients with positive leading coefficient such that(?)ai|f(z)for some integerz.Suppose that each ai>0 for 1 ? i ? m,then there exists an integer N0 ? N(k,m,n)such that N ? N0 implying that each k-coloring of the integers of[N]= ?1,2,...N}contains pairwise distinct monochromatic x1,x2,...,xm ?[N]satisfying a1x1+a2x2...+amxm? f(Z).In 1975,Szemeredi obtained a famed theorem by investigating the arithmetic progressions of the set of integers with positive density:For a real number ?>0 and an integer k ? 1,there exists a positive integer N so that,if the subset S of[N]satisfies |S| ? ?N then S contains a ?-term arithmetic progression.In Chapter 3 of this dissertation,we get the following results by deliberating the density on the integers.Theorem 2.Suppose t ? 2 is an integer,0<(?)<1/4 is a real number and N>0 is a sufficiently large integer.If B(?)C[N]and |B|<log N/(21og 1/e),then there exists a set A(?):C[N]such that |A|>(1/t-(1/t + 1/4)(?))[N/t]and a1+a2+...+at(?)B for ai,A(1? i?t).Theorem 3.If t>2 is an integer,?>1 is a real number and B =?b1,b2,...,bk,...} is a strictly increasing infinite sequence of positive integers with infk=1,2,...bk+1/bk ? ?,then there exists a strictly increasing infinite sequence A={a1,a2,...,ak...} of positive integers such that liminfN??[N]|/N ?exp {-(log 3/(log ?)+ 1)log(12t)} and ax1 + ax2 +… axt(?)B.