w(k;3) > 2^{k (log^* k)/4} for large k, so the three-color van der Waerden number grows super-exponentially.
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6 Pith papers cite this work. Polarity classification is still indexing.
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UNVERDICTED 6representative citing papers
New lower bounds on g_3(n) via central trinomial coefficients and general exponential lower bounds for g_k(n) (k≥4), paired with upper exponential rates from digit constructions on nearly-regular graphs.
Measurable sets in [0,R]² avoiding upward right triangles of area 1/2 satisfy |A| = O_c(R²/(log R)^c) for c<1/4 with Ω(R log R) example; for fixed-area triangles the bound sharpens to c<1/2 using a hyperbolic trilinear smoothing inequality and scale induction.
Dense subsets of [N]^n contain configurations x, x + r^{m1}e1, ..., x + r^{mn}en for any fixed n and increasing exponents m_i, with density threshold (log N)^{-c}.
Non-existence and lower-tail probabilities in the critical regime for hypergraph edge counts are approximated by the Bethe free energy at the unique fixed point of a Belief Propagation operator under structural conditions on the hypergraph.
f(n²,n) ≥ n + (1/√2 + o(1))√n and f(p²,p) ≤ 2p − (√(2/3) − o(1))√(p/log p) for large primes p.
citing papers explorer
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Three-color van der Waerden numbers grow super-exponentially
w(k;3) > 2^{k (log^* k)/4} for large k, so the three-color van der Waerden number grows super-exponentially.
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Arithmetic Progression-Free Subset-Sum Sets
New lower bounds on g_3(n) via central trinomial coefficients and general exponential lower bounds for g_k(n) (k≥4), paired with upper exponential rates from digit constructions on nearly-regular graphs.
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On hyperbolic corners and unit-area triangles in planar sets of large measure
Measurable sets in [0,R]² avoiding upward right triangles of area 1/2 satisfy |A| = O_c(R²/(log R)^c) for c<1/4 with Ω(R log R) example; for fixed-area triangles the bound sharpens to c<1/2 using a hyperbolic trilinear smoothing inequality and scale induction.
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A multidimensional Szemer\'{e}di theorem in integers
Dense subsets of [N]^n contain configurations x, x + r^{m1}e1, ..., x + r^{mn}en for any fixed n and increasing exponents m_i, with density threshold (log N)^{-c}.
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Non-existence probabilities and lower tails in the critical regime via Belief Propagation
Non-existence and lower-tail probabilities in the critical regime for hypergraph edge counts are approximated by the Bethe free energy at the unique fixed point of a Belief Propagation operator under structural conditions on the hypergraph.
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Hitting Arithmetic Progressions at the Square-Root Scale
f(n²,n) ≥ n + (1/√2 + o(1))√n and f(p²,p) ≤ 2p − (√(2/3) − o(1))√(p/log p) for large primes p.