An old conjecture of Erd\H os and Szemer\'edi states that if $A$ is a finite set of integers then the sum-set or the product-set should be large. The sum-set of $A$ is defined as $A+A=\{a+b | a,b \in A\}$ and the product set is $A\cdot A=\{ab | a,b \in A\}.$ Erd\H os and Szemer\'edi conjectured that the sum-set or the product set is almost quadratic in the size of $A,$ i.e. $\max (|A+A|,|A\cdot A|)\geq c|A|^{2-\delta}$ for any positive $\delta$. I proved earlier that $\max (|A+A|,|A\cdot A|)\geq c|A|^{14/11}/\log{|A|},$ for any finite set of complex numbers, $A.$ In this talk we improve the bound further for sets of real numbers.