Union of shadows

Our aim in this note is to show that if <f>A</f> is a family of <f>(<NU>k</NU>/3</fr>)</f> 3-sets then there are at least <f>(<fr style="bl" shape="case" ALIGN="C"><NU>k</NU>/4)</f> 3-sets then there are at least <f>(<NU>k</NU>/4)</f> 4-sets that may be written as unions of two 2-sets in the shadow of <f>A</f>. This is the first non-trivial case of a more general conjecture about unions of shadows. [Copyright &y& Elsevier]