We let denote the crossing number of a graph .

Conjecture   Every graph with satisfies .

We say that a set is -universal if every vertex planar graph can be drawn in the plane so that each vertex maps to a distinct point in , and all edges are (non-intersecting) straight line segments.

Question   Does there exist an -universal set of size ?

Let denote the diagonal Ramsey number.

Conjecture   exists.

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.

Let be a class of finite relational structures. We denote by the number of structures in over the labeled set . For any class definable in monadic second-order logic with unary and binary relation symbols, Specker and Blatter showed that, for every , the function is ultimately periodic modulo .

An directed graph is -diregular if every vertex has indegree and outdegree at least .

Conjecture   For , every -diregular oriented graph on at most vertices has a Hamilton cycle.

Say that a family of graphs is -bounded if there exists a function so that every satisfies .

Conjecture   For every fixed tree , the family of graphs with no induced subgraph isomorphic to is -bounded.

Basic Question: Given any positive integer n, can any convex polygon be partitioned into n convex pieces so that all pieces have the same area and same perimeter?

Definitions: Define a Fair Partition of a polygon as a partition of it into a finite number of pieces so that every piece has both the same area and the same perimeter. Further, if all the resulting pieces are convex, call it a Convex Fair Partition.

Questions: 1. (Rephrasing the above 'basic' question) Given any positive integer n, can any convex polygon be convex fair partitioned into n pieces?

2. If the answer to the above is "Not always'', how does one decide the possibility of such a partition for a given convex polygon and a given n? And if fair convex partition is allowed by a specific convex polygon for a give n, how does one find the optimal convex fair partition that minimizes the total length of the cut segments?

3. Finally, what could one say about higher dimensional analogs of this question?

Conjecture: The authors tend to believe that the answer to the above 'basic' question is "yes". In other words they guess: Every convex polygon allows a convex fair partition into n pieces for any n

To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total perimeter of pieces is (1) maximized (2) minimized.

Remark: It appears maximizing the total perimeter is the easier problem.

Let be a matroid of rank , and for let be the number of closed sets of rank .

Conjecture   is unimodal.

Conjecture   is log-concave.

Conjecture   Let be the complete funcoid corresponding to the usual topology on extended real line . Let be the order on this set. Then is a complete funcoid.

Proposition   It is easy to prove that is the infinitely small right neighborhood filter of point .

If proved true, the conjecture then can be generalized to a wider class of posets.

Conjecture   The poset of multifuncoids of the form is for every sets and :

\item atomic; \item atomistic.

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

Conjecture   (solved) If then for every funcoid and atomic f.o. and on the source and destination of correspondingly.

A stronger conjecture:

Conjecture   If then for every funcoid and , .