What is the minimum amount of points that can win a frame of snooker?

Players are penalized between 4 points and 7 points for committing a foul.

http://www.usapoolhalls.com/billiards/snooker/

A foul can occur for numerous reasons, such as hitting a colour first when the player was attempting to hit a red, potting the cue ball, or failing to escape from "a snooker" (a situation where the previous player finished their turn leaving the cue ball in a position where the object ball cannot be hit directly). Points gained from a foul vary from a minimum of 4 to a maximum of 7 if the black ball is involved.

https://en.wikipedia.org/wiki/Snooker

Snooker is played using a cue and snooker balls: one white cue ball, 15 red balls worth one point each, and six balls of different colours yellow (2 points), green (3), brown (4), blue (5), pink (6) and black (7).

https://en.wikipedia.org/wiki/Snooker

When a red ball is knocked into a pocket ("potted") using the cue ball, 1 point is scored and the player must then pot a coloured ball of his choice. To score the minimum points, therefore, each red ball must be followed by a yellow giving a score of 15 x (1 + 2) or 45 points. The player then clears the colours to score 2 + 3 + 4 + 5 + 6 + 7 points or 27 points. This gives a grand total of 72. The opponent must play correctly and not commit any fouls, which would award more points to the first player.

The minimum score when the opponent concedes because he cannot match the first player is not 40 - 0. If the first player pots 12 reds each followed a yellow he will score 12 x (1 + 2) = 36. He then pots another red but does not pot the yellow. This gives him 37 points. Play passes to his opponent who then has 2 reds (2pts) which he can follow by 2 blacks (14pts) then all the colours (27pts) for a score of 43 which wins the game. The first player must pot 14 reds and 13 yellows for a score of 40pts. His opponent can then only score 1pt for the remaining red, 7pts for its associated black then 27pts for the colours, to a total of 35pts.

Again, since the opponent can concede at any time the winning score could theoretically be 0 - 0 if he concedes without a ball being potted.

If all 15 reds are potted by player "A" without pocketing a colour, that is 15 pts. If player "B" then proceeds to pocket Yellow, Green, Brown, Blue and Pink in order they will score 20 pts. Then if player "A" pots the final Black, that will give them 22 pts.

Player A: 22 pts
Player B: 20 pts

I'm sure i'm right, as there is a minimum of 42 pts to be "won" The Colours add up to 27, and the reds add up to 15. Total 42.

My mistake was in thinking a potted red must be followed by a potted yellow (lowest scoring colour) but of course the yellow need not be potted at all!

With 42 points available (as you correctly point out) a draw would be 21 - 21 so the minimum winning score must be 22 - 20.

A little more consideration of the scoring system is required I believe the answer is 22-13. Because :- player A gets 8 reds = 8 points player B 7 reds = 7 points A then pots yellow = 2 points, green = 3 points, brown = 4 points and blue = 5 points puts him on a total of 22 points. Player B then pots pink bringing him to 13 points but has foolishly left him more than 7 points behind, end of frame.

12 points to 0 is the minimum.

Player A misses the reds at break off three times giving 12 points in fouls and by the 3 miss rule forfeits the frame.

http://www.rulesofsnooker.com/understanding-the-three-miss-rule.html

You CAN actually get 22-20, instead of 22-13.

A pots all 15 reds.
B pots pots Yellow(2), Green(3), Brown(4), Blue(5), and Pink(6).

The current score would be 15-20. Making Black(7) the deciding factor.

Which obviously, Player A pots it and wins the frame with score 22-20.

If we apply the rule that "Frame Ends when the frame ball is scored", then we have these exceptions:
22-7 is the minimum score with black and pink (13points) left
(8red+first 4 colored ball vs 7red)
21-1 is the minimum score with black pink blue (18pts) left.
(14red+first 3 colored ball vs 1 red)