Question

The edges of K_n are colored red and blue in such a way that a red edge is in at most one red triangle. Show that there is a subgraph K_k with k >= sqrt(2n) that contains no red triangle.

Answer #1

Let the edges of K7 be colored with the colors red
and blue. Show that there are at least four subgraphs K3 with
all
three edges the same color (monochromatic triangles). Also
show
that equality can occur.

Let the edges of K7 be colored with the colors red
and blue. Show that there are at least four subgraphs K3
with all three edges the same color (monochromatic triangles). Also
show that equality can occur.

Suppose that each object in an n-object list L
is colored either red or blue. Give an efficient EREW algorithm to
form two lists from the objects in L: one consisting of
the blue objects and one consisting of the red objects.

Suppose that each object in an n-object list L is colored either
red or blue.
Give an efficient EREW algorithm to form two lists from the objects
in L: one consisting of the blue objects and
one consisting of the red objects.

Assume that each point on a circle is colored either red or
blue. Prove that there are three points, say P, Q, R, that are
colored the same and where at least two of the three distances
between these points, d(P, Q), d(P, R), and d(Q, R), are equal.

One colored chip - red blue, or green is selected at random and
a fair die is rolled.
a) Use the counting principle to determine the number if sample
points in the sample space.
b) Construct a tree diagram illustrating all the possible
outcomes and list the sample space.

There is a box with two colored balls, red balls and blue balls.
The probability of selecting a red ball is 10%. The balls are
independent of each other, and after each trial of choosing a ball,
you put it back in the box.
1.) What is the probability your first red ball that you get
will be on your 3rd or 5th trial?
2.) What is the probability your first red ball will be on an
even trial? (Trial...

An urn contains 10 chips, of which 5 are blue and 5 are red.
Randomly select 5 chips, one at a time without replacement. Let X
be the absolute difference between the numbers of blue and red
chips that have been selected.
a) Find the pmf of X. Show your work.
b) What value of X is most likely?

In an ornamental plant, silver-colored flowers (s) are recessive
to blue-colored flowers (S), red leaves (r) are recessive to green
leaves (R), and long stems (l) are recessive to short stems (L).
All three genes involved are located on the same chromosome. Two
true-breeding strains were crossed to produce an F1 plant. Then,
this F1 plant was test-crossed to a plant with silver flowers, red
leaves, and long stems, to produce the following progeny:
progeny phenotype
count of progeny
silver...

Urn A contains two red balls and eight blue balls. Urn B
contains two red balls and ten green balls. Six balls are drawn
from urn A and four are drawn from urn B; in each case, each ball
is replaced before the next one is drawn. What is the most likely
number of blue balls to be drawn? What is the most likely number of
green balls to be drawn?

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