In this episode, a deadly virus is spreading across Los
Angeles.

A graph is defined as two sets V and E. V is any non-empty set
and is called the vertex set. Elements of E are pairs of elements
of V, and E is unsurprisingly called the edge set. Graph theory
is the study of properties of these graphs with applications ranging
from computers to social networking to epidemeology. Suppose you
draw a hundred points on a piece of paper. You pick one special
point v and play connect the dots, draw edges from one vertex to the
next. Then you start back at v and do it again. You repeat
the process a dozen or so times. You then ask your friend to
figure out which vertex you started from. This is essentially the
problem the FBI and CDC are working on, to find the first infected
person(s), only there are hundreds of thousands of people all linked
together in the contaminated area, so the problem is vastly more
complex.

The problem (and beauty) of graph theory is that many problems can
be simply stated and sometimes even relatively simply solved, but from
a practical standpoint are all but impossible. In other words,
one can prove the existence of something with relative ease, but giving
an example of one could be extremely difficult. One such problem
is the Ramsey coloring problem. We call a graph complete if every
two vertices are joined by an edge. *K*_{n}
denotes the complete graph on n vertices. A 2-colored graph is a
graph G together with a partition of the vertex set into two
subsets. Suppose we agree to call the two colors red and
blue Given a 2-colored complete graph G, is there a number R(r)
so that if G consists of more than R vertices then G has a complete
monochromatic subgraph on r vertices? Here monochromatic means
that all the vertices in the subgraph are all in one of the two
partitioned subsets. Ramsey proved that yes, there is.
Further, he extended the resul to the following question: is there a
number R(r,s) so that if there are more than R vertices in a 2-colored
complete graph G then there is either a complete red subgraph on r
vertices or a complete blue subgraph on s vertices. Again the
answer is yes. The smallest number R(r,s) is called a Ramsey
number. Note that when r=s, this new problem corresponds to the
original problem. Even for very small numbers, the Ramsey numbers
are not known precisely. R(5,5), for example, is not known
precisely, although it is known to be between 43 and 49. The
prolific twentieth-century mathematician Erdős expresses the
computational difficulty of the problem with the quip, "Imagine an
alien force, vastly more powerful than us landing on
Earth and demanding the value of R(5, 5) or they will destroy our
planet. In that case, we should marshal all our computers and all our
mathematicians and attempt to find the value. But suppose, instead,
that they asked for R(6, 6), we should attempt to destroy the aliens."

Proofs of very simple theorems are quite complicated. The Four
Color Theorem is an example of such a problem. A planar graph is
a graph that can be drawn on a plane without having its edges
cross. Another way to think of such a graph is by taking the
plane, cutting it up into regions, and drawing a graph by putting a
vertex inside each region and then drawing edges between vertices whose
regions border one another. The Four Color Theorem says that
given a planar graph, one can color the vertices with 4 colors so that
no adjacent vertices are of the same color. The question comes
from the desire to color a map of nations so that no bordering nations
are the same color. This is slightly different since nations are
not generally continguous. For example, during the colonial times
of the Great Britain, that nation had numerous colonies across the
globe. The Four Color Theorem is known for having one of the
least elegant proofs of all time. The original proof of the
theorem reduced the problem to almost 2000 cases which were then
handled using a computer.

Try computing R(3,3).

A bipartite graph is a graph whose vertex set can be partitioned
into two sets so that vertices in each partition are only joined by
edges to vertices in the other partition. A graph is connected if
there is a path between any two vertices (where path has the intuitive
meaning). A minimally connected graph is called a tree.
Prove that every tree is bipartite. A cycle is exactly what it
sounds like. Prove that a cycle with an even number of vertices
is bipartite.

A complete bipartite graph is a bipartite graph with the property
that each vertex in one partition is joined to every vertex in the
other partition, i.e. it is a bipartite graph with the most possible
edges. *K*_{m,n}_{}
denotes the complete bipartite graph partitioned into sets of m and n
vertices. There is an interesting relationship between a graph
being planar and the graphs *K*_{3,3} and *K*_{5}. Learn more
about that.

When Charlie visits Larry at the restaurant, Larry makes a very
nerdy joke. The punch line is that it's "just a run of the
Yang-Mills black hole." Yang-Mills (and the mass-gap problem) is
a problem of mathematical physics. Yang-Mills is the underpinning
of elementary particle theory that has made many correct and testable
predictions, but the mathematics is not at the level of rigor required
of mathematicians. One of the major problems is establishing that
there is a Yang-Mills theory with the property that quantum particles
have positive mass. This mass-gap problem is one of the
$1,000,000 Clay Institute Millenium Problems.