abstract
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Fans and bundles in the graph of pairwise sums and products

Lorenz Halbeisen

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Let *G* be the graph on the vertex-set the positive integers
*N*, with *n* joined to *m* if *n* and *m*
are distinct and for some *x,y*
in *N* we have *x+y = n* and *x y = m*. A pair of
triangles sharing an edge and containing three consecutive
numbers is called a 2-fan, and three triangles on 5 numbers
having one number in common and containing four consecutive numbers is
called a 3-fan. It will be shown that *G* contains 3-fans,
infinitely many 2-fans and even arbitrarily large *bundles* of
triangles sharing an edge. Finally, it will be shown that the
chromatic number of *G* is at least 4.