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From: TCL on 9 May 2010 13:08
Characterize numbers a,b,c,d,e such that there exist five
sets A_i, i=1,..,5 satisfying

|A_i|=a for all i;

|A_i \cap A_j|=b, for all i=/ j;

|A_i \cap A_j \cap A_k|=c for all distinct i,j,k

and so on.

Of course two neccessary conditions on these numbers are that

5a-10b+10c-5d+e must be positive, and that a,b,c,d,e must be

nonincreasing.

(Here |A| denotes the cardinality of A.)
From: TCL on 9 May 2010 14:28
On May 9, 1:08 pm, TCL <tl...(a)cox.net> wrote:
> Characterize numbers a,b,c,d,e such that there exist five
> sets A_i, i=1,..,5 satisfying
>
> |A_i|=a for all i;
>
> |A_i \cap A_j|=b, for all i=/ j;
>
> |A_i \cap A_j \cap A_k|=c for all distinct i,j,k
>
> and so on.
>
> Of course two neccessary conditions on these numbers are that
>
> 5a-10b+10c-5d+e must be positive, and that a,b,c,d,e must be
>
> nonincreasing.
>
> (Here |A| denotes the cardinality of A.)

Here are more necessary conditions:
a+c >= 2b;
a+5c >= 3b+d;
a+e >= 4(b+c+d).
From: TCL on 9 May 2010 14:36
On May 9, 2:28 pm, TCL <tl...(a)cox.net> wrote:
> On May 9, 1:08 pm, TCL <tl...(a)cox.net> wrote:
>
>
>
>
>
> > Characterize numbers a,b,c,d,e such that there exist five
> > sets A_i, i=1,..,5 satisfying
>
> > |A_i|=a for all i;
>
> > |A_i \cap A_j|=b, for all i=/ j;
>
> > |A_i \cap A_j \cap A_k|=c for all distinct i,j,k
>
> > and so on.
>
> > Of course two neccessary conditions on these numbers are that
>
> > 5a-10b+10c-5d+e must be positive, and that a,b,c,d,e must be
>
> > nonincreasing.
>
> > (Here |A| denotes the cardinality of A.)
>
> Here are more necessary conditions:
> a+c >= 2b;
> a+5c >= 3b+d;
> a+e >= 4(b+c+d).- Hide quoted text -
>
> - Show quoted text -

Correction: The last one should be a+4c+e >= 4(b+d).
From: TCL on 9 May 2010 15:23
On May 9, 1:08 pm, TCL <tl...(a)cox.net> wrote:
> Characterize numbers a,b,c,d,e such that there exist five
> sets A_i, i=1,..,5 satisfying
>
> |A_i|=a for all i;
>
> |A_i \cap A_j|=b, for all i=/ j;
>
> |A_i \cap A_j \cap A_k|=c for all distinct i,j,k
>
> and so on.
>
> Of course two neccessary conditions on these numbers are that
>
> 5a-10b+10c-5d+e must be positive, and that a,b,c,d,e must be
>
> nonincreasing.
>
> (Here |A| denotes the cardinality of A.)

I think the following characterize these numbers, but have yet to
prove it:

a \ge b \ge c \ge d \ge e \ge 0;
a-2b+c \ge 0;
a-3b+3c-d \ge 0;
a-4b+6c-4d+e \ge 0

(Here \ge stands for greater than or equal to.)
From: TCL on 9 May 2010 16:22
On May 9, 3:23 pm, TCL <tl...(a)cox.net> wrote:
> On May 9, 1:08 pm, TCL <tl...(a)cox.net> wrote:
>
>
>
>
>
> > Characterize numbers a,b,c,d,e such that there exist five
> > sets A_i, i=1,..,5 satisfying
>
> > |A_i|=a for all i;
>
> > |A_i \cap A_j|=b, for all i=/ j;
>
> > |A_i \cap A_j \cap A_k|=c for all distinct i,j,k
>
> > and so on.
>
> > Of course two neccessary conditions on these numbers are that
>
> > 5a-10b+10c-5d+e must be positive, and that a,b,c,d,e must be
>
> > nonincreasing.
>
> > (Here |A| denotes the cardinality of A.)
>
> I think the following characterize these numbers, but have yet to
> prove it:
>
> a \ge b \ge c \ge d \ge e \ge 0;
> a-2b+c \ge 0;
> a-3b+3c-d \ge 0;
> a-4b+6c-4d+e \ge 0
>
> (Here \ge stands for greater than or equal to.)- Hide quoted text -
>
> - Show quoted text -

I should also add

b-2c+d \ge 0;
c-2d+e \ge 0;
b-3c+3d-e \ge 0.
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