## 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.

#### Permutations and permutation groups

Infinite groups
• H1: "Original Collatz problem"
• H8: Proportion of derangements
• H10: Orbits on pairs
• H28: Exponential constants for orbit counts
• H32: N-free graphs and denominator identities
• H40(S): Cycle index of product
• H58: Pointwise product of orbit-counting sequences
• H60: Automorphism groups of Henson's graphs
• H70: A conjugacy class representation of the random graph?
• H84:Countable B-groups
• H107: the random graph and highly transitive groups
• P2(part S): An integral domain from a permutation group
• B28: Abelian automorphism groups of generic n-tuples 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
• H27(S): q-analogues of fixed points and cycles
• H30: Automorphisms inverting k elements
• H31: Automorphism group transitive on non-commuting 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
• H65: A near coincidence in symmetric groups
• H68, P26: Rank and subrank of primitive permutation groups
• 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 2-closed 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
• H43: Sets with constant distance in PSL(2,q)
• B9: Invariant set of random permutation
• B12, B13: Permutations with prescribed distances
• B14: Sharply s-transitive sets
• B17: "Geometric sets" of permutations
• BCC13.19: Permutations with few distances
Semigroups
• H102: Random synchronization
• H106: Synchronizing non-uniform maps
Miscellaneous
• H75: Multipermutation solutions of the Yang--Baxter equation

#### Designs and configurations

Designs
• H11: "Anti-biplanes"
• 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 Z4-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: Block-transitive designs
• BCC16.19: Semi-Latin 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
• 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

#### Graph theory

Enumerative
• H2(S): Trees of height k
• H72: Orbital flow roots
• H103: Graphs with few subgraphs
• P10: Graphs with prescribed automorphism group (see also B25 and B26)
• BCC12.5: Edge-density of infinite graphs
• BCC15.8: Counting classes of graphs
Structural
• H26(S): Pigeonhole property
• H46: Homomorphism-homogeneous graphs
• H49: Positive and negative eigenvalues
• H60: Automorphism groups of Henson's graphs
• H67: Circular altitude of a graph
• H69: Core-transitive 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

#### Miscellaneous problems

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 lambda-roots
• 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: Sum-free sets in the square and cube
• H66(S): Generalised parking numbers
• H76(S): VC-dimension and strength of Gray map images
• BCC12.6(S): Intersecting families
• BCC13.14: Cyclic shifts of binary words
• BCC13.24(S): Sum-free sets containing 2
• BCC14.10(S): Even and odd permutations
• BCC14.11(S): How many sum-free sets?
Algebra
• H88: Linearity of order-preserving maps
• H89: Endomorphisms and partial isomorphisms of groups
Other
• H7(S): Rolling triangular prism
• H27(S): q-analogues of fixed points and cycles
• H32: N-free 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): Measure-preserving 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–Ryll-Nardzewski–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 Hadamard-like and conference-like matrices

#### Algorithms and complexity

• 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