A covering design, or covering, is a family of -subsets, called blocks, chosen from a -set, such that each -subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by .
Problem Find a closed form, recurrence, or better bounds for . Find a procedure for constructing minimal coverings.
Conjecture Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.
In particular, this implies:
Conjecture Twin Prime Conjecture: There are an infinite number of twin primes.
For every finite multiplicative group , let () denote the smallest integer so that every sequence of elements of has a subsequence of length (length ) which has product equal to 1 in some order.
Conjecture Let be a simple graph with vertices and list chromatic number . Suppose that and each vertex of is assigned a list of colors. Then at least vertices of can be colored from these lists.
Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an grid. The first player (if any) to occupy a set of cells having no two cells in the same row or column is the winner. What is the outcome of the game given optimal play?
Begin with the generating function for unrestricted partitions:
(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...
Now change some of the plus signs to minus signs. The resulting series will have coefficients congruent, mod 2, to the coefficients of the generating series for unrestricted partitions. I conjecture that the signs may be chosen such that all the coefficients of the series are either 1, -1, or zero.
Problem Is there a problem that can be computed by a Turing machine in polynomial space and unbounded time but not in polynomial time? More formally, does P = PSPACE?
Conjecture If is a bridgelesscubic graph, then there exist 6 perfect matchings of with the property that every edge of is contained in exactly two of .
A friendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.
Problem Is it true that for every , all but finitely many -regular graphs have friendly partitions?
Conjecture If is a simple graph which is the union of pairwise edge-disjoint complete graphs, each of which has vertices, then the chromatic number of is .
Consider a set of great circles on a sphere with no three circles meeting at a point. The arrangement graph of has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points. So this arrangement graph is 4-regular and planar.
Conjecture Every arrangement graph of a set of great circles is -colourable.
Conjecture Let and are monovalued, entirely defined funcoids with . Then there exists a pointfree funcoid such that (for every filter on ) (The join operation is taken on the lattice of filters with reversed order.)
A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.
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be an abelian group of odd order and let 
satisfy 
. Then the elements of 
and 
may be ordered 
and 
so that the sums 
are pairwise distinct. 
covering design, or covering, is a family of 
-subsets, called blocks, chosen from a 
-set, such that each 
-subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by 
.
. Find a procedure for constructing minimal coverings. 
vertices, then 
for 
. 
be a set of ![\[m(A) = - \min_x \sum_{a \in A} \cos(ax).\]](/files/tex/7a772ebd0b7fd98eeab7fda3a5e9674b1dc4c984.png)
What is 
? 
(
) denote the smallest integer 
so that every sequence of 
(length 
) which has product equal to 1 in some order. 
for every finite group 
exist a 
. Suppose that 
and each vertex of 
vertices of 
grid. The first player (if any) to occupy a set of 
has distinct subset sums if distinct subsets of 
have distinct sums.
so that 
whenever 
has distinct subset sums. 
structurally stable diffeomorphism is hyperbolic. 
-
of 
so that 
is 
-edge-connected and 
is 
-edge-connected? 
. Does the fragment 
have the finite model property? 
has chromatic number at most 
. 
of 
, all but finitely many 
is the minimum number 
for which there exists a polytope 
with 
with 
.
? 
be an integer and let 
denote the least integer 
-colourable. 
and 
are monovalued, entirely defined funcoids with 
. Then there exists a pointfree funcoid 
such that (for every filter 
on 
(The join operation is taken on the lattice of filters with reversed order.) 
does there exists a 
such that every graph with average degree smaller than 
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