Conjecture For every fixed graph , there exists a constant , so that every graph without an induced subgraph isomorphic to contains either a clique or an independent set of size .
Conjecture \item If is a countable connected graph then its third power is hamiltonian. \item If is a 2-connected countable graph then its square is hamiltonian.
Conjecture There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number .
The number appears once in Pascal's triangle, appears twice, appears three times, and appears times. There are infinite families of numbers known to appear times. The only number known to appear times is . It is not known whether any number appears more than times. The conjectured upper bound could be ; Singmaster thought it might be or . See Singmaster's conjecture.
Problem (2) Find a composite or which divides both (see Fermat pseudoprime) and the Fibonacci number (see Lucas pseudoprime), or prove that there is no such .
Conjecture Let be an integer. For every integer , there exists an integer such that for every digraph , either has a pairwise-disjoint directed cycles of length at least , or there exists a set of at most vertices such that has no directed cycles of length at least .
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?
Problem What is the maximum number of colours needed to colour countries such that no two countries sharing a common border have the same colour in the case where each country consists of one region on earth and one region on the moon ?
Problem () Find a sufficient condition for a straight -stage graph to be rearrangeable. In particular, what about a straight uniform graph?
Conjecture () Let be a simple regular ordered -stage graph. Suppose that the graph is externally connected, for some . Then the graph is rearrangeable.
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.
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always rational? 
, there exists a constant 
, so that every graph 
without an induced subgraph isomorphic to 
. 
. Does the fragment 
have the finite model property? 
is the minimum number 
for which there exists a polytope 
with 
with 
.
, a convex polygon on 
? 
denote the pebbling number of a graph 
. 
-diregular if every vertex has indegree and outdegree at least 
, every 
-diregular oriented graph on at most 
vertices has a Hamilton cycle. 
. 
appears once in Pascal's triangle, 
appears twice, 
appears three times, and 
appears 
times. There are infinite families of numbers known to appear 
times is 
. It is not known whether any number appears more than 
. See 
or 
which divides both 
(see 
(see 
-regular graph, where 
, then 
has chromatic number at most 
. 
be an integer. For every integer 
, there exists an integer 
such that for every digraph 
, either 
, or there exists a set 
of at most 
vertices such that 
has no directed cycles of length at least 
contains every 2-regular graph on 
vertices as a minor. 
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? 
and a function 
such that if 
, then every saturated 
has cardinality at least 
? 
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 
![$ p \in [0,1] $](/files/tex/1d076f7332523eb59bd7deae19667f83f0a3b6e0.png)
, we let 
denote a subgraph of 
.
so that 
? 
) Find a sufficient condition for a straight 
) Let 
be a simple regular ordered 
is externally connected, for some 
. Then the graph 
is rearrangeable. 
covering design, or covering, is a family of 
-set, such that each 
.
. Find a procedure for constructing minimal coverings. 
, then 
cycles of length 
. 
is for every sets 
and 
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