Problem Two players start at a distance of 2 on an (undirected) line (so, neither player knows the direction of the other) and both move at a maximum speed of 1. What is the infimum expected meeting time (first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy?
Problem Let be natural numbers with . It follows from the pigeon-hole principle that there exist distinct subsets with . Is it possible to find such a pair in polynomial time?
Conjecture For all there is an integer such that every digraph of minimum outdegree at least contains a subdivision of a transitive tournament of order .
Let be a non-empty finite set. Given a partition of , the stabilizer of , denoted , is the group formed by all permutations of preserving each block of .
Problem () Find a sufficient condition for a sequence of partitions of to be complete, i.e. such that the product of their stabilizers is equal to the whole symmetric group on . In particular, what about completeness of the sequence , given a partition of and a permutation of ?
Conjecture (Beneš) Let be a uniform partition of and be a permutation of such that . Suppose that the set is transitive, for some integer . Then
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 Let be a graph, a countable end of , and an infinite set of pairwise disjoint -rays in . Prove that there is a set of pairwise disjoint -rays that devours such that the set of starting vertices of rays in equals the set of starting vertices of rays in .
Conjecture Suppose with is a connected cubic graph admitting a -edge coloring. Then there is an edge such that the cubic graph homeomorphic to has a -edge coloring.
Conjecture Let be an -uniform-partite hypergraph. If is the maximum number of pairwise disjoint edges in , and is the size of the smallest set of vertices which meets every edge, then .
The zeroes of the Riemann zeta function that are inside the Critical Strip (i.e. the vertical strip of the complex plane where the real part of the complex variable is in ]0;1[), are actually located on the Critical line ( the vertical line of the complex plane with real part equal to 1/2)
Conjecture The following statements are equivalent for every endofuncoid and a set : \item is connected regarding . \item For every there exists a totally ordered set such that , , and for every partion of into two sets , such that , we have .
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.
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(first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy? 
for every endo-
? 
be an indexed family of sets.
for 
.
on the lattice 
of all finite unions of products.
a bijection from hyperfuncoids 
defining the inverse bijection? If not, characterize the image of the function 
of complements of elements of the set 
-cube into 
be natural numbers with 
. It follows from the pigeon-hole principle that there exist distinct subsets 
with 
. Is it possible to find such a pair 
in polynomial time? 
there is an integer 
such that every digraph of minimum outdegree at least 
. 
for principal funcoid 
of funcoids of appropriate sources and destinations. 
, there exists a Fibonacci number divisible by 
, 
does not divide 
where 
is the Legendre symbol. 
be a non-empty finite set. Given a partition 
of 
, is the group formed by all permutations of 
.
) Find a sufficient condition for a sequence of partitions 
of 
is equal to the whole symmetric group 
on 
, given a partition 
of 
be a uniform partition of 
be a permutation of 
. Suppose that the set 
is transitive, for some integer 
. Then 
be an abelian group. Is the 
be a graph, 
a countable end of 
an infinite set of pairwise disjoint 
of pairwise disjoint 
is a connected cubic graph admitting a 
-edge coloring. Then there is an edge 
such that the cubic graph homeomorphic to 
has a 
be an 
-
is the maximum number of pairwise disjoint edges in 
is the size of the smallest set of vertices which meets every edge, then 
. 
is connected regarding 
.
and a set 
:
there exists a totally ordered set 
such that 
, 
, and for every partion 
of 
into two sets 
, 
such that 
, we have ![$ X \mathrel{[ \mu]^{\ast}} Y $](/files/tex/0ef560be389646efd1fdde5ebc9afc9ac98ee64e.png)
. 
. Suppose that 
and each vertex of 
colors. Then at least 
vertices of 
there is 
such that, for every large 
and 
. 
, what is the smallest integer 
such that 
? 
be the unique integer (with respect to a fixed 
) such that
is a prime iff ![$$ \displaystyle \sum_{i=1}^{\left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor} r_i = \left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor $$](/files/tex/99af565f4cc4d3bab11eb3fbf54f78626678d484.png)
vertices which contains no complete 
hyperedges. 
vertices which contains no complete 
hyperedges. 
is for every sets 
and 
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