SUCH THAT X+Y AND X·Y ARE MONOCHROMATIC. SUCH THAT X+Y AND X·Y AR...

2, such that x+y and x·y are monochromatic.Let us state this problem in terms of graphs: Let G

^×

₊

= (N, E) be the graph on thevertex-set the positive integers N, with (n, m) ∈ E if n 6= m and for some x, y ∈ N wehave n = x+y and m = x·y. Notice that for all n ∈ N, (n, n+ 1) ∈ E (this is justbecause n = 1·n and n+ 1 = 1 +n). If we colour N, then a monochromatic edgeofG

^×

₊

is an edge (n, m) such that n has the same colour as m. Now, the question reads asfollows: If we colour Nwith finitely many colours, does G

^×

₊

has a monochromatic edge?If there would be a 2-colouring ofNsuch thatG

^×

₊

has just finitely many monochromaticedges, then we could easily construct a finite colouring of N such that no edge of G

^×

₊

ismonochromatic. But it is not hard to show that G

^×

₊

has arbitrarily “large” triangles,and therefore, for any 2-colouring of N, G

^×

₊

has arbitrarily “long” monochromatic edges.In fact, for any positive integer m there are x

0

, y

0

, x

1

, y

1

, x

2

, y

2

∈ N such that m <min{x

₀

, y

0

, x

1

, y

1

, x

2

, y

2

} and x

0

+y

0

=x

1

+y

1

, x

1

·y

1

=x

2

+y

2

and x

0

·y

0

=x

2

·y

2

. Tosee this, fix some positive integer m and let a ∈ N be such that a > m. Let h be anypositive integer and define x

0

, x

1

, x

2

, and n as follows: x

0

= 2ha

²

, x

1

= a, x

2

=ha, andn= 4ha

²

−h−a. Further, let y

0

=n−x

0

= 2ha

²

−h−a, y

1

=n−x

1

= 4ha

²

−h−2a,and y

2

=x

1

·y

1

−x

2

= 2a(2ha

²

−h−a). Now, by definition we have x

0

+y

0

=x

1

+y

1

and x

1

·y

1

=x

2

+y

2

, and in addition we also getx

0

·y

0

= 2ha

²

(2ha

²

−h−a) =x

2

·y

2

,and m <min{x

₀

, y

0

, x

1

, y

1

, x

2

, y

2

}.A pair of triangles sharing an edge (i.e., a K

4

with an edge deleted) and containing3 consecutive numbers is called a 2-fan, and three triangles on 5 numbers having onenumber in common and containing 4 consecutive numbers is called a 3-fan. In the sequelit will be shown that G

^×

₊

contains 3-fans, infinitely many 2-fans and even arbitrarily large“bundles” of triangles sharing an edge, and an algorithm is provided to generate such≥4.“bundles”. Further, it will be shown thatχ G

^×

₊

Acknowledgement: I would like to thank Imre Leader who lighted my interest in thistopic by his excellent talk at the15th September Meeting of theCumann Matamaitice nah ´Eireannin Cork (Ireland). Further, I would like to thank Stephanie Halbeisen for manyhints and fruitful discussions, and the referee for numerous extremely helpful suggestionsand remarks on an earlier version of this paper.

2 Fans

For a positive integer `, an `-fan is a subgraph ofG

^×

₊

of the following type:. . .

^//

n−`n

^//

L

L

n−1

^//

=

=

n−2

^//

L

L

=

=

=

%%

L

yyrrr

rrr

rrr

rrr

rrr

rrr

rrr

rr

mwhere an arrow fromntom indicates that there arex, y ∈Nwithx+y=nandx·y=mrespectively.The following result provides a characterization of`-fans.Theorem <sub>2.1.</sub> For `, n, m ∈ N with n > `, the integers n, n−1, . . . , n−`, m are thevertices of an `-fan if and only if there are positive integers a and b with b ≤ a suchthat n = a +b + 2ab +`, m = a(a + 1)b(b + 1), and for each t ∈ {1,2, . . . , ` − 1},p(a+b+t+ 1)

²

+ 4abt is an integer.Proof. Sufficiency: For each i∈ {0,1, . . . , `} let(a+b+`−i)

²

+ 4ab(`−1−i)k

i

= (n−i)−p