Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Group Theory
akman_sissokho_2025_steiner_coset_partitions_groups/: Constructs equal-index Steiner coset partitions and gives structural nonexistence results relevant to, but not resolving, Erdős Problem 274.
akman_sissokho_2025_transversal_coset_partitions_groups/: Proves the Herzog--Schönheim conjecture for partitions using at most seven distinct subgroups and develops direct-product and parallelism constructions that still repeat subgroup indices.
akman_sissokho_2026_steiner_coset_partitions_five_mutually_commuting_subgroups/: Classifies Steiner coset partitions from five distinct proper mutually commuting subgroups, leaving only a recursive four-subgroup construction.
akman_sissokho_2026_steiner_coset_partitions_groups_erratum/: Corrects the scope of a nonexistence claim in the 2025 paper's abstract and repairs its defining notation for a_J.
beker_2025_most_probable_order_random_permutation/: Shows the largest probability that a random permutation of n letters has a given order is asymptotic to 1/n, and identifies that order exactly for all large n.
berger_et_al_1986_herzog_schonheim_conjecture_finite_nilpotent_groups/: Proves the Herzog-Schönheim conjecture for finite nilpotent groups by encoding cosets as prime-power product sets and compressing their union.
berger_et_al_1987_remark_multiplicity_partition_group_into_cosets/: Proves repeated-index multiplicity bounds for exact coset partitions of finite pyramidal groups, including all finite supersolvable groups.
erdos_1965_probabilistic_methods_group_theory/: Shows that about log base two of n random elements of an abelian group of order n let every element be written as a subset sum.
erdos_1976_probabilistic_methods_group_theory/: Shows that for almost all choices of k random elements of a finite abelian group every element has nearly the average number of subset-sum representations.
erdos_1978_new_results_probabilistic_group_theory/: Gives a Poisson law for the number of group elements with a prescribed count of subset-sum representations from random generators.
gao_geroldinger_2006_zero_sum_problems_finite_abelian_groups/: Surveys zero-sum sequence invariants in finite abelian groups, including Davenport constants, inverse structure, subsequence counts, cross numbers, and critical numbers.
garonzi_margolis_2025_herzog_schonheim_conjecture_simple_symmetric_groups/: Proves the Herzog-Schönheim conjecture for simple and symmetric groups via reciprocal sums of distinct subgroup indices, with asymptotics and limits of the method.
ginosar_schnabel_2011_prime_factorization_conditions_multiplicities_coset_partitions_groups/: Proves the Herzog–Schönheim conjecture for several families of finite groups using prime-factorization bounds and intersecting support hypergraphs.
girard_2008_inverse_zero_sum_algebraic_invariants/: Studies the maximal cross number of long zero-sumfree sequences in finite abelian groups and proves the inverse conjecture for finite cyclic groups, finite abelian p-groups, and finite abelian groups of rank two.
itabe_2026_herzog_schonheim_conjecture_coset_partitions_at_most_seventeen_cells/: An unreviewed computer-assisted proof candidate, with a complete retained Lean 4 release, claiming that any counterexample requires at least eighteen cells.
korec_znam_1977_disjoint_covering_groups_cosets/: Source record and research digest.
kriz_2013_maximal_cross_number_unique_factorization_sequences/: Bounds the maximal cross number of unique factorization indexed multisets over finite abelian groups, proving the Gao–Wang formula for several families and asymptotically approaching it in broad classes.
lam_leung_2000_vanishing_sums_roots_unity/: Determines the possible weights of vanishing sums of m-th roots of unity and analyzes minimal relations through integral group rings of cyclic groups.
lettl_sun_2008_covers_abelian_groups_cosets/: Records Lettl and Sun's index and Mycielski-function bounds for exact and minimal multiple covers of abelian groups by cosets.
menon_2026_two_questions_g_harmonic_tuples/: Gives an explicit non-integer-harmonic coset partition of A5, with repeated indices, and rules out a proposed five-subgroup counterexample pattern.
nagy_pach_2026_group_ring_identity_alon_jaeger_tarsi/: Formulates a group-ring implication for nonsingular matrices over prime fields and proves that it would imply the Alon–Jaeger–Tarsi non-vanishing-vector conjecture.
neumann_1976_problem_paul_erdos_groups/: Answers Erdős's question affirmatively: the groups whose non-commuting graph has no infinite complete subgraph are exactly the groups whose center has finite index, and then the complete subgraphs have bounded size.
pyber_1987_number_pairwise_non_commuting_elements_index/: Proves that a group with at most n pairwise non-commuting elements has centre of index at most c^n, answering questions of B. H. Neumann and Erdős.
roneydougal_2025_subgroups_symmetric_groups_enumeration_asymptotic_properties/: Proves the symmetric group on n points has 2^(n^2/16 + o(n^2)) subgroups, confirming a 1993 conjecture of Pyber.
sun_1990_finite_coverings_groups/: Gives lower bounds and extremal characterizations for partitions of groups into cosets of finite-index subnormal subgroups.
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