Rainbow Solutions of a Linear Equation with Coefficients in Z/pZ

Let p be a prime, n is an element of Z(+) and w is an element of (0, 1). Given a colouring chi : Z/pZ -> {1, 2, & mldr;, n} and a linear equation L : a(1)x(1 )+ a(2)x(2 )+ & ctdot; + a(n)x(n )= b with a(1), a(2), & mldr;, a(n )is an element of (Z/pZ)(& lowast;) and b is an element of Z/pZ, we denote by R(chi, L) the family of vectors (b(1), b(2), & mldr;,b(n)) is an element of (Z/pZ)(n) such that a(1)b(1 )+ a(2)b(2 )+ & ctdot; + a(n)b(n )= b ad chi(-1)(i)boolean AND{b(1), b(2), & mldr;, b(n)} not equal & empty; for each i is an element of {1, 2,& mldr;, n}. In this paper it is shown that there exists a constant c = c(w, n) > 0 with the following property: if min 1(<= i <= n)|chi(-1)(i)| >= wp + 1 >> p(3/4) and if there exist coefficients a(i) and a(j) such that a(i )is not an element of{+/- a(j)}, then |R(chi,L)| >= cp(n-1).