On the number of monochromatic Schur triples

Let <f>S={1,2,…,n}</f>, and let <f>S=S1∪S2</f> be a partition of <f>S</f> in two disjoint subsets. A triple <f>(a,b,c)</f>, <f>a,b,c∈S</f>, is called a Schur triple if <f>a+b=c</f>. If in addition <f>a</f>, <f>b</f>, and <f>c</f> all lie in the same subset <f>Si</f> of <f>S</f>, we call the triple <f>(a,b,c)</f> monochromatic. In this paper we give a simple proof that the minimal number of monochromatic Schur triples is asymptotic to <f>n2/11</f>. We also show that the number of monochromatic Schur triples modulo <f>n</f> equals <f>n2−3&z.sfnc;S1&z.sfnc;&z.sfnc;S2&z.sfnc;</f>. [Copyright &y& Elsevier]