[Chris]

I believe this is it.
First update yum

yum update

then try mysql-server-5.0

yum install mysql-server-5.0

if that does not work use

yum search mysql-server

thats should give you a list of packages that match mysql-server, mysql 5.0 is what i use.

I do not use CentOS But i have so i know a little. Mysql is the same on all OS's

~$ mysql -u root mysql

then
since it will be a new install you will need to do this first.
add password to root account

mysql> UPDATE user SET password=PASSWORD('desired_password')
->WHERE User='root';

you should see

Query OK, 2 rows affected (0.00 sec)
rows matched: 2 Changed: 2 warnings: 0

Then since its new do

mysql>DELETE FROM user WHERE user = '';

mysql>FLUSH PRIVILEGES;

mysql>DROP DATABASE test;

mysql>exit