The problem in the proof is when we add the vertex y to the x1,...,xt. 
The vertices x1,...,xt have the property of a strong system 
but you do not know anything about the vertex y.
<p>
The Füredi-Tuza paper have also the version with k+1 and they give 
an example that it is tight.

<a href="http://dx.doi.org/10.1016/S0012-365X(85)80009-1">  
http://dx.doi.org/10.1016/S0012-365X(85)80009-1 </a>
<p>
<a href="http://www.inf.ethz.ch/personal/zuphilip/"> Philipp Zumstein</a>