Combinatorics
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Paper Counterexamples to Sidorenko’s conjecture and the forcing conjecture
Disproves Sidorenko's conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing…
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Paper Superexponential van der Waerden numbers
Resolves Erdős's superexponential-growth question for van der Waerden numbers. If Wr(k) is the least interval length forcing a monochromatic k-term progression in every r-coloring, then $`Wr(k)\gt k^{ck\lfloor\log2…
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Paper Classification of finite Euclidean Ramsey configurations
Classifies finite point configurations that occur monochromatically, at their original scale, in every finite coloring of sufficiently high-dimensional Euclidean space. The characterization is an algebraic condition…
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Paper Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds
Proves Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. Quantitatively, for each fixed k ≥ 3, every subset of…
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Paper The Euclidean plane cannot be colored with five colors
Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes. This advances the Hadwiger–Nelson problem: together with the classical…
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Paper Planar distinct distances and unit-distance bounds
Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but o(n) points of any n-point planar set determine at least n^{1-\varepsilon} distinct nonzero distances. A complementary theorem…
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Paper The higher-dimensional Erdős distinct-distances conjecture
For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝd determine at least cdn^{2/d} distinct distances, with cd\gt 0 depending only on dimension. This matches the integer-grid order and resolves the…
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