Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Combinatorics
L113_butterfly_triangle_degree_sum/: Every order n at least ten trillion admits a graph above the Mantel edge threshold whose triangle degree sums are below 183n/125.
L153_johnson_pair_moments/: A positive-semidefinite, entrywise nonnegative two-point relaxation for coloring the whole k-layer of a 2k-set with k+1 colors is feasible for every even k, including all remaining prime cases of Problem 835.
L187_complete_cloning_obstruction/: Fano caps obstruct complete cloning for each seed, and seven parity rows exclude every nontrivial clone expansion that would retain both the local seven-edge property and a ratio above three quarters.
L238_normalized_core_near_clique_bound/: An r-set with h missing edges in a normalized two-supported core satisfies r at most a(h+1), excluding the one-hole fold pattern when r exceeds 2a.
L25_bipartite_four_choosability/: Palette minimization, subset covering and color incidences give the lower bound n(4)>=26 for bipartite list coloring.
L26_design_size_multiplicity/: A nontrivial pairwise balanced design needs at least n blocks, and near-square-root minimum block size forces too few sizes for an O(sqrt(n)) multiplicity cap.
L27_blocking_set_cap/: Incidence moments exclude blocking sets with at most three points per line in every projective plane of order at least five.
L78_bipartite_order_26_four_choosable/: Every bipartite graph with at most 26 vertices is 4-choosable, so n(4) is at least 27; a small integer weighting of palette subsets excludes order 26.
These claims concern finite set systems, designs, incidence structures and graph colourings. A list colouring question on a complete bipartite graph is a question about two families of palette subsets, and the claims here state their conventions for lists, palettes and side splits with the proposition itself.