
1. (a) Show that in any 2coloring of the edges of the complete
graph on 6 vertices, there are at least two monochromatic triangles.
1. (b) Show that in any 2coloring of the edges of a complete graph on
7 vertices, there are at least 4 monochromatic triangles.
2. A student has 37 days to prepare for an exam. She knows she will
require no more than 60 hours of study. She wishes to study at
least 1 hour per day. Show that no matter how she schedules her
study time (in an integer number of hours per day), there is a
succession of days during which she will have studied exactly 13 hours.
3. Prove that no matter how the integers 1, 2, ..., 12 are
positioned around a circle, some set of three consecutive integers
sums to at least 19.
4. Prove Shur's Theorem.
5a Find a lower bound on S(3,4).
5b Bonus: Show that the Schur number s(3) = 14.
Q1 Find the number of distinct bracelets of 5 beads made from
yellow, blue and white beads. Two bracelets are indistinguishable
if the rotation of one will yield another (no flips allowed).
Q2 The six faces of a cube are to be painted with 6 different
colors, each face with a distinct color. In how many ways can this
be done?
Q3: How many ways are there to 3color the vertices of a wheel $W_5$?
Bonus: Determine the number of directed graphs on 3 unlabeled vertices
using Polya techniques.