Problem pages index
The list of problems which have appeared on my homepage and were subsequently
"put out to grass" is growing rather long, so I have tried to index them, to
make them easier to use. (There are now 165 problems here, though some are
duplicates and some have been solved.) At the same time I have integrated my
collections of permutation group problems into the main list. Comments welcome!
The notation (S) means that the problem has been solved;
the text includes either a reference to the solution, or a sketch of how
it is done. Of course, even a solved problem might be worth generalising!
Some problems appear in more than one class.
Index
In this index, numbers (Hn) refer to problems taken from my homepage;
numbers (Bn) refer to problems from my article "Permutations" in the
Budapest conference Paul Erdős and his Mathematics;
numbers (Pn) are other problems on permutations and permutation groups;
and numbers (BCCm.n) to problems I have presented at the
British Combinatorial Conference. The problems live in separate files:
homepage problems,
problems from "Permutations",
permutation group problems, and
BCC problems. (Note: the complete listing of
BCC problems is also available.)
The problems are classified as follows.
 Infinite groups

 H1: "Original Collatz problem"
 H8: Proportion of derangements
 H10: Orbits on pairs
 H28: Exponential constants for orbit counts
 H32: Nfree graphs and denominator
identities
 H40(S): Cycle index of product
 H58: Pointwise product of orbitcounting
sequences
 H60: Automorphism groups of Henson's graphs
 H70: A conjugacy class representation of
the random graph?
 H84:Countable Bgroups
 H107: the random graph and highly
transitive groups
 P2(part S): An integral domain from a
permutation group
 B28: Abelian automorphism
groups of generic ntuples of orders
 Finite groups

 H20(S): Squares outside corefree subgroup
 H22(S): Groups of type {k}
 H23(S): Groups with all elements outside a
normal subgroup having prime order
 H25: Partitions and permutations
(see also B30)
 H27(S): qanalogues of fixed points
and cycles
 H30: Automorphisms inverting k
elements
 H31: Automorphism group transitive on
noncommuting pairs
 H34: Tutte polynomial and cycle index for
IBIS groups
 H36: Antichain in subgroup lattice
 H37: Embedding Boolean lattice in subgroup
lattice
 H48(S): Matrix algebra from Frobenius group
 H52: Base size and separation number
 H54: Bijective proof of Dixon's theorem
 H64: Covers of symmetric groups (see
also BCC21.13)
 H65: A near coincidence in symmetric groups
 H68, P26:
Rank and subrank of primitive permutation groups
 H77: Derangements as short words (see also
P27)
 H80: Universal families of permutation
groups
 H81: Does "spreading" imply QI?
 H100: Generating a group by coset representatives
 P1(S?): Isbell's problem
 P4: Semiregular elements in 2closed groups
 P5: Matroids from IBIS groups
 P6: Finding small generating sets
 P7(S): Base size in almost simple primitive
groups
 P8: Permutation groups and association
schemes
 P9: A derangement problem
 P10: Graphs with prescribed automorphism
group (see also B25 and
B26)
 P11: Parker vectors
 B1: Mixing time for the
"Burnside process"
 B4: A derangement problem
 B7: Derangements of prime order
 B15: Sharp groups
 B19: Pyber's conjecture on
base size
 B20: Greedy base size
 B22: Groups meeting the strong
Maillet bound
 B23: Groups with greedy bases
of constant size
 B27(S?): Hypergraphs with
prescribed automorphism group
 Sets

 H41: Covering radius for sets of
permutations (see also B29 and
BCC19.12)
 H43: Sets with constant distance in
PSL(2,q)
 B9: Invariant set of random
permutation
 B12,
B13: Permutations with
prescribed distances
 B14: Sharply
stransitive sets
 B17: "Geometric sets" of
permutations
 BCC13.19: Permutations with few
distances
 Semigroups

 H102: Random synchronization
 H106: Synchronizing nonuniform maps
 Miscellaneous

 H75: Multipermutation solutions of the
YangBaxter equation
 Designs

 H11: "Antibiplanes"
 H14: Designs and partitions
 H15: Random Latin squares
 H21(S): Symmetric graph designs
 H57(S): Balanced Sudoku
 H78(S): Optimality and Tutte polynomial
 H79: A very weak Hadamard conjecture
 H86: The strength of the Gray map image of
a Z_{4}code
 H87: Intricacy of Latin squares
 H97: Covering a design with blocks
 B10,
B11: Random Latin squares
 BCC13.17: A generalisation of affine
designs
 BCC14.8: The rows of a Latin square
 BCC14.13: Blocktransitive designs
 BCC16.19: SemiLatin squares which
are partial linear spaces
 BCC17.4: An extremal problem related
to biplanes
 BCC18.9: A Markov chain for Steiner
triple systems
 Finite geometry

 H24: Separation in inversive planes
 H38: Derangements in affine planes
 BCC12.4: Generalised quadrangles
 BCC13.12: Permutations of projective
space
 BCC18.5: Projective space analogues
of Steiner systems
 Association schemes

 H33: Jordan schemes
 H48(S): Matrix algebra from Frobenius group
 P8: Permutation groups and association
schemes
 BCC16.16: Abnormal strongly regular
graphs
 Matroids

 H34: Tutte polynomial and cycle index for
IBIS groups
 P5: Matroids from IBIS groups
 B23: Groups with greedy bases
of constant size
 BCC16.23: Representing orthogonal
matroids
 BCC17.19(S): Covering radius and
Tutte polynomial
 Enumerative

 H2(S): Trees of height k
 H3,
H5: Probability of connectedness (see also
BCC15.8)
 H59: Orbital chromatic roots (see also
BCC21.12)
 H72: Orbital flow roots
 H103: Graphs with few subgraphs
 P10: Graphs with prescribed automorphism
group (see also B25 and
B26)
 BCC12.5: Edgedensity of infinite
graphs
 BCC15.8: Counting classes of graphs
 Structural

 H9(S): "Orthogonal" edgecolourings
(see also BCC16.15(S))
 H26(S): Pigeonhole property
 H46: Homomorphismhomogeneous graphs
 H49: Positive and negative eigenvalues
 H50(S): Semiregular automorphisms (see
also BCC17.12)
 H60: Automorphism groups of Henson's graphs
 H67: Circular altitude of a graph
 H69: Coretransitive graphs
 H70: A conjugacy class representation of
the random graph?
 H73: Algebraic properties of chromatic
roots
 H85(S): The power graph of a finite group
 H91(S): Infinite hulls
 H92: The power graph of a finite magma
 B27(S?): Hypergraphs with
prescribed automorphism group
 BCC21.2: The row space of an
adjacency matrix
 H101: Chromatic number of power graph
 Number theory

 H16: Sums of two prime powers
 H17: Sums of consecutive primes
 H18: Fibonacci matters
 H19: An iteration on primes
 H29: Sums of reciprocals
 H35,
H42(S): Primitive lambdaroots
 H45: Sequences realisable by fixed points
of powers
 H53: Determinants of Legendre symbols
 H71: Orders of finite simple groups
 H83: Decomposing the group of units by
arithmetic progressions
 H90(S): A lim inf problem
 H93: A universal sequence from the primes?
 H105: Subgroups of the multiplicative group
 H108(S): A Diophantine approximation problem
 H109: More on Problem 19
 BCC12.3: Two Diophantine equations
 BCC12.7: Forbidding divisibility
 BCC20.10: A determinant problem
 Combinatorics

 H4(S): Necklaces and polynomials
 H6(S): Rapidly increasing sequences
 H12(S): Central binomial coefficients
 H13(S): Coprime polynomials
 H39: Sumfree sets in the square and cube
 H41: Covering radius for sets of
permutations (see also B29 and
BCC19.12)
 H66(S): Generalised parking numbers
 H76(S): VCdimension and strength of Gray
map images
 BCC12.6(S): Intersecting families
 BCC13.14: Cyclic shifts of binary
words
 BCC13.24(S): Sumfree sets
containing 2
 BCC14.10(S): Even and odd permutations
 BCC14.11(S): How many sumfree sets?
 Algebra

 H88: Linearity of orderpreserving maps
 H89: Endomorphisms and partial isomorphisms
of groups
 Other

 H7(S): Rolling triangular prism
 H27(S): qanalogues of fixed points
and cycles
 H32: Nfree graphs and denominator
identities
 H44: Extending partial permutations
 H47: Growth of Stirling transforms
 H51: Möbius number of polyhedral
groups and connection number of root lattices (see also
BCC20.16)
 H54: Bijective proof of Dixon's theorem
 H55: Shadows of permutations and score
sequences of tournaments
 H56(S): Easy Sudoku
 H57(S): Balanced Sudoku
 H61(S): Measurepreserving transformations
of [0,1]
 H62(S): Computational complexity of
generalised Sudoku
 H63: Hilbert series of an algebra related
to multiplicative functions
 H74: Network flow where channels silt up
 H82(S): Bulgarian solitaire
 H94: Engeler–RyllNardzewski–Svenonius for monoids?
 B9: Invariant set of random
permutation
 H98: closed abelian groups of isometries
of Urysohn space
 H99: synchronization properties of random
transformations
 H104: determinants of Hadamardlike and
conferencelike matrices
 H44: Extending partial permutations
 H56(S): Easy Sudoku
 H62(S): Computational complexity of
generalised Sudoku
 H95: An undecidable random graph?
 H96: Optimisation where you pay for
computer time
 P6: Finding small generating sets
(see also B24)
Peter J. Cameron
p.j.cameron(AT)qmul.ac.uk
26 December 2012