big
I get very nervous before a big race.

close, tight
long-distance
10-mile, 24-hour, 7-lap, etc.
gruelling/grueling, hard, tough
relay, road
men's, women's
boat, yacht, etc.
bicycle, bike (informal), car, motor (esp. BrE), motorbike (BrE), motorcycle, etc.
dog, horse, etc.
drag, turf (esp. AmE)
play-off (esp. AmE)

have, hold, organize
Let's have a race!

be in, compete in, do (informal), enter, go in for, run in, take part in
Is she running in the big race on Saturday?

drop out of, pull out of, withdraw (sth) from
He dropped out of the race with a pulled muscle after two laps.
She had to pull out of the race at the last minute.

lead, win
I was leading the race until the half-way point.
She has won the race for the last five years.

lose
come first in, come second in, etc.
beat sb/sth in
fix
People are saying that the race was fixed.

throw (= to lose deliberately)
He was paid $10 000 to throw the race.

be held, be run, take place

meet (esp. AmE), meeting (BrE)
The horse was withdrawn from today's race meeting with an injured leg.

winner
official, organizer

in a/the race
I'm not in this race.

out of a/the race
race against, race with
the race against the Danish team

race between
A close race between the top two boats is expected.

race over
a race over two miles

close, tight
two-horse
Although there are five candidates, realistically it is a two-horse race. (figurative)

congressional, gubernatorial (AmE), presidential, Senate
arms, space
the halting of the nuclear arms race

be in, be involved in, join
The rival TV companies are in a race to bring out the first film drama of his life.

be ahead in, lead, win
Who will win the race for the White House?

be left behind in, lag behind in, lose

in a/the race
He is lagging behind in the race for the presidency.

race between
a race between the developing countries

race for
the race for nuclear supremacy

race with
the race with their rivals

human
alien
mixed
a child of mixed race

Irish, Jewish, Latin, etc.
master
the ideology of the master race

relations
Immigration and race relations were key political issues at the time.

equality
discrimination, preference, prejudice
riot

among races
The disease is more common among European races.

between races
greater understanding between nations and races

from a race
children from all races and religions

of a race
people of different races and cultures

regardless of race
We can all work together, regardless of race.

race of
The Amazons were a race of female warriors.

on the basis of race, on (the) grounds of race
There is no excuse for discrimination on the grounds of race.