Conjecture Suppose runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1. Then for any given runner, there is a time at which that runner is distance at least (along the track) away from every other runner.
Problem What is the largest connected planar graph of minimum degree 3 which has everywhere positive combinatorial curvature, but is not a prism or antiprism?
Conjecture If is the adjacency matrix of a -regular graph, then there is a symmetric signing of (i.e. replace some entries by ) so that the resulting matrix has all eigenvalues of magnitude at most .
Conjecture There exists an integer such that every -arc-strong digraph with specified vertices and contains an out-branching rooted at and an in-branching rooted at which are arc-disjoint.
Conjecture If a finite set of unit balls in is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease.
Conjecture Let is a -separable (the same as for symmetric transitive) compact funcoid and is a uniform space (reflexive, symmetric, and transitive endoreloid) such that . Then .
The main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.
The direct proof may be constructed by correcting all errors an omissions in this draft article.
Direct proof could be better because with it we would get a little more general statement like this:
Conjecture Let be a -separable compact reflexive symmetric funcoid and be a reloid such that \item ; \item .
Conjecture For every graph without a bridge, there is a flow .
Conjecture There exists a map so that antipodal points of receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero.
Let be a hypergraph. A strongly maximal matching is a matching so that for every matching . A strongly minimal cover is a (vertex) cover so that for every cover .
Conjecture If is a (possibly infinite) hypergraph in which all edges have size for some integer , then has a strongly maximal matching and a strongly minimal cover.
Conjecture For every , there exists an integer such that if is a digraph whose arcs are colored with colors, then has a set which is the union of stables sets so that every vertex has a monochromatic path to some vertex in .
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,
. 
and every positive integer 
, there exists 
so that every simple 
-regular graph 
has a 
with 
. 
the following inequality holds 
is for every sets 
and 
:
is the minimum number 
for which there exists a polytope 
with 
with 
.
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such that the list chromatic number of any bipartite graph 
is at most 
.
denote the pebbling number of a graph 
. 
(along the track) away from every other runner. 
be an integer and let 
denote the least integer 
is the adjacency matrix of a 
-regular graph, then there is a symmetric signing of 
entries by 
) so that the resulting matrix has all eigenvalues of magnitude at most 
. 
be an integer and let 
. Then either 
minor for some 
or 
with specified vertices 
and 
contains an out-branching rooted at 
is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease. 
is a 
-separable (the same as 
for symmetric transitive) compact funcoid and 
is a uniform space (reflexive, symmetric, and transitive endoreloid) such that 
. Then 
. 
. 
. 
.
so that antipodal points of 
receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero. 
so that 
for every matching 
. A strongly minimal cover is a (vertex) cover 
so that 
for every cover 
. 
for some integer 
, then 
cycles of length 
. 
such that 
or prove that such integers do not exist. 
are matroids on 
and 
for every partition 
of 
with 
which is independent in every 
. 
such that if 
set which is the union of 
be the flow polynomial of a graph 
equals the number of 
for every 2-edge-connected graph 
into 
so that the graph induced by 
is acyclic for 
. 
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