How many points to qualify?

The best possible case I could found is ( by symmetry )

Let a,b,c,d,e,f,g,h be the 8 teams

a ends in draw with b,c,d...b ends in draw with a,d,c ....c ends in draw with a,b,d......d ends in draw with a,b,c ......rest all matches they win.......giving all the 4 the same score of 11

For a team to surely get in the top 4 it must get a number of points , k , such that for that respective number of points it will be certain that there aren't any other distinct 4 teams which will get the same or a greater number of points therefore it can be taught that there must be a score of k points which if a team has then there is impossible for other 4 teams to get in any way of distributing the total points to get the same or a bigger score. Therefore , in order to understand this it can be taught of it in the terms of how large must be k such that it affects the scores of the other teams in such a way that there will be impossible for 4 teams to get the number of points equal or greater with k , this being the "mechanism" of point distribution among teams in which to inquire (by thinking how does a score affect the other scores). Firstly observe that for a number of points which a team can get there will be a number of different configurations regarding the number of wins , loses and equal matches a team had with the others and that if the interest of the a team will be for a fixed number of points which the team will have to maximize the difference between it and other teams then it should rather prefer a configuration of wins which doesn't give any number of points to the other teams than a configuration which has equal matches since it will give a number of points to the other teams. If that is the case then for a team it would be preferable to achieve the same score without having any draw matches which means that in the worst case for a team it will have a configuration for some number of points of matches that are draws and as in this inquiry there should be an understanding of the number of points a team must get to be assured it belongs to the top 4 then there should be taken into account the worst possible case since if there are for a score some certain configurations that do not work it is not assured for the team that enters in the top 4. Therefore it can be presupposed that there are equal matches between the teams and it can be started from a equal distribution of points among teams. Observe that each team plays against 7 others and therefore are 7! = 28 games played where in one of the two possible way there will always be in each game distributed 2 points therefore there are 56 points in total that are distributed among the teams therefore starting from an equal configuration of point distribution it may be assumed that every team has 7 points. Starting from this distribution and taking a team as the one which must get into the top 4 it can be said that it can be made a win by taking a maximum of 1 point each of the team and a loss for that team from which the points where taken and in order to understand better the possible distribution of points consider it abstractly. Now , for a team the number of points for that team will be 7 plus the number of points r taken from any other distinct r teams which , if the team must be assured that gets in the top 4 should lead to a number of points such that there is not possible in any way of distributing the points for other 4 teams to get the same score as that team. Supposing this is not the case for some r and thinking abstractly at the distribution of points among teams it can be therefore applied in a subtle way the pigeonhole principle by the fact that for that team to take r points from some r teams it must be that it must also take r be large enough to make it impossible for any other 4 teams to have a bigger score. If r will not be large enough to affect more than 3 teams it would mean that there will always be 4 teams which can also take the same number of points the first team took and therefore it means that it can equal the number of points of that team leading to a contradiction which implies therefore that for 4 teams for a number of r points taken by the that team to not be capable to exceed that number it must also affect some of these 4 teams and it can be shown that for affecting 1 from this teams therefore in this case for r = 4 that team will also need to affect in order to achieve an equal score to the team which needs to be in the top 4 to affect the same number of teams , therefore also 4 which leads to an impossible case therefore the minimum score being for any configuration possible 4 + 7 = 11 points.