**Problem 1** (*Barrington, Beigel* and *Rudich*
[2]) Does there exist a polynomial *P* in *n* variables, with
integer coefficients, of degree ,
which weakly
represents the *n*-variable OR function modulo 6? (Recall, that this
means that
,
and
for any
.)

If the answer is yes for some
and the polynomials
are explicitly constructed, then our method yields explicit
Ramsey-graphs on

vertices, with no complete subgraph and no independent set of size

For symmetric polynomials, *Barrington, Beigel* and *Rudich*
[2] have shown that the degree is
.

Showing only the *existence* of polynomials, weakly representing
the OR function with degree ,
would also have considerable
theoretical interest, since this result would imply the existence of
larger set-systems in Theorem 1.2. Here we should also mention
that the best lower bound is due to *G. Tardos * and *Barrington* [9]. They proved that if the modulus *m* has *r*>1
different prime divisors, then every polynomial, weakly representing
the function OR_{n} modulo *m*, has degree at least

**Problem 2** Does there exist a quadratic polynomial *P*
in *n* variables, with integer coefficients, which weakly represents
the *n*-variable OR function modulo
,
where both
and
are ? If the answer is yes, then
combining this *P* and the polynomial of *Barrington, Beigel* and
*Rudich* [2], we would obtain a polynomial, satisfying the
requirements of Problem 1.

**Problem 3**
It remains an open question whether, for a fixed positive integer *m*,
a better than exponential (
upper bound holds
for the size of set-systems satisfying that the size of each set
is divisible by *m* while the sizes of their pairwise intersections
are not divisible by *m*.

This problem is open even for *m*=6.
Our main result shows that if *m* is not a prime power then
no polynomial upper bound (*O*(*n*^{c})) holds. (If *m* is a prime power
then a polynomial upper bound holds by Frankl - Wilson 1.1.)

**Problem 4**
If in Problem 3 we assume additionally that the sizes of the pairwise
intersections occupy only two residue classes mod *m* then there
may even be a polynomial upper bound (perhaps *O*(*n*^{2})), yet we are
not aware of any better-than-exponential upper bound even for this case.
This, too, is open for *m*=6.

**Acknowledgments.** The author wishes to thank
*Zoltán Király* and *David Mix Barrington* for fruitful
discussions and to *Péter Frankl* for valuable comments and suggestions.
The author is especially grateful to *Laci Babai* for
numerous helpful remarks and suggestions. The author acknowledges
the support of grants OTKA T030059, FKFP 0607/1999, AKP 97-56 2,1.