Figure 1

How many people must you have at dinner to ensure that
there are a subset of 3 people who all either
mutual acquaintances, or mutual strangers?
We can model this using graph theory;
for background see the Fun Fact Six Degrees of Separation.
Let people be vertices, and draw a red edge between two
people if they know each other, and a blue edge if they
do not. So every pair of people are connected by either
a red or blue edge.
The question then becomes, what is the least number of
vertices for which every complete redblue graph
on those vertices has
either a red or blue triangle?
Answer: 6 people! First we show that 6 people are enough:
take any vertex, Fred. Fred has 5 edges emanating from
him; at least 3 are the same color.
Suppose without loss of generality it is red. If any of
those 3 people that Fred knows know each other, then
we have a red triangle! If none of
those 3 know each other, we have a blue triangle!
Now, using redblue graphs, can you draw a redblue
complete graph on 5 vertices with no blue nor red triangle?
(This shows that 5 people are not enough.)
See the Figure for a solution (it appears after several
seconds).
Presentation Suggestions:
Bring colored chalk into the class;
draw pictures as you are giving the proof.
It doesn't matter if students don't follow the
exact argument in class they will probably go home
and want to think about it anyways but they will
draw the pictures you draw and it will help them
construct the argument later.
The Math Behind the Fact:
Generalizations of this problem, involving the
least number of people you have to invite to ensure
subgraphs of other types, fall under the category of
something called Ramsey Theory.
How to Cite this Page:
Su, Francis E., et al. "Dinner Party Problem."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
