number-theory Apr 25, 2026 Significance 10/100 Registry: site confirmed
Prior state unknown → proved
VibeMathed records this result as “Erdős Problem #888.” The registry entry and named primary source contain the available statement and scope.
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number-theory Jul 13, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
If A A A is a forbidden-divisor set with ∣ A ∩ [ 1 , x ] ∣ = o ( x ) |A \cap [1,x]| = o(\sqrt{x}) ∣ A ∩ [ 1 , x ] ∣ = o ( x ) and B = { b 1 < b 2 < ⋯ } B = \{b_1 < b_2 < \cdots\} B = { b 1 < b 2 < ⋯ } the sifted set, must x − 1 ∑ b i < x ( b i + 1 − b i ) 2 x^{-1} \sum_{b_i < x} (b_{i+1} - b_i)^2 x − 1 ∑ b i < x ( b i + 1 − b i ) 2 converge to a finite limit?
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combinatorics Aug 19, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
Every finite simple connected graph G G G with
∣ V ( G ) ∣ = 2 d + 1 , diam ( G ) = d ≥ 3
|V(G)|=2d+1,\qquad \operatorname{diam}(G)=d\ge 3
∣ V ( G ) ∣ = 2 d + 1 , diam ( G ) = d ≥ 3
satisfies
W ( G ) ≤ W ( C 2 d + 1 ) = ( 2 d + 1 ) d ( d + 1 ) 2 .
W(G)\le W(C_{2d+1})
=\frac{(2d+1)d(d+1)}2.
W ( G ) ≤ W ( C 2 d + 1 ) = 2 ( 2 d + 1 ) d ( d + 1 ) .
The claimed equality cases are exactly C 2 d + 1 C_{2d+1} C 2 d + 1 for every d ≥ 3 d\ge3 d ≥ 3 , the double star D 2 , 3 D_{2,3} D 2 , 3 when d = 3 d=3 d = 3 , and the nine-vertex tree T 1 , 2 , 2 = S ( 2 , 3 , 3 ) T_{1,2,2}=S(2,3,3) T 1 , 2 , 2 = S ( 2 , 3 , 3 ) when d = 4 d=4 d = 4 .
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number-theory Jan 10, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → disproved
VibeMathed records this result as “Erdős Problem #397.” The registry entry and named primary source contain the available statement and scope.
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number-theory Apr 16, 2026 Significance 10/100 Registry: site confirmed
Prior state unknown → proved
VibeMathed records this result as “Erdős Problem #1217.” The registry entry and named primary source contain the available statement and scope.
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probability-statistics May 15, 2026 Significance 10/100 Registry: expert verified
Prior state unknown → proved
For the switch-walk-switch walk on Z 2 ≀ Z \mathbb{Z}_2 \wr \mathbb{Z} Z 2 ≀ Z started at ( 0 , 0 ) (0,0) ( 0 , 0 ) and ( 0 , 2 ) (0,2) ( 0 , 2 ) , prove ∥ P t x − P t y ∥ T V ≍ t − 1 / 2 \|P_t^x - P_t^y\|_{TV} \asymp t^{-1/2} ∥ P t x − P t y ∥ T V ≍ t − 1/2 .
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algorithms-optimization Jul 18, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
Prim-Dijkstra routing interpolates between a minimum spanning tree and a shortest-path tree, and has been used and improved in VLSI physical design since the early 1990s, but the complexity of the terminal-only Manhattan decision problem was never settled. It is weakly NP-complete. A continuous cost-radius tradeoff with a balanced ( 2 , 2 ) (2,2) ( 2 , 2 ) guarantee accompanies the classification.
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number-theory Jul 25, 2026 Significance 10/100 Registry: contested
Prior state unknown → proved
For the least k k k at which the small-prime part of ( n k ) \binom{n}{k} ( k n ) exceeds n 2 n^2 n 2 , how large can f ( n ) f(n) f ( n ) be?
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analysis May 17, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
order of magnitude determined; the exact asymptotic constant remains open
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number-theory Jul 13, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
upper bound improved to f(n) ≤ 14n^{3/7} with an explicit logarithmic lower bound; matching bounds remain open
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combinatorics Jun 9, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → disproved
VibeMathed records this result as “Erdős Problem #619.” The registry entry and named primary source contain the available statement and scope.
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algorithms-optimization Jun 12, 2026 Significance 10/100 Registry: site confirmed
Prior state unknown → disproved
On the basis of experiments up to 5000 nodes, Papamanthou and Tollis conjectured a relation between the longest paths produced by their MaxSTN and MinSTN algorithms for s t st s t -orientations of biconnected graphs. A counterexample refutes it.
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analysis Feb 1, 2026 Significance 10/100 Registry: expert verified
Prior state unknown → disproved
capacity alone does not determine the invariant; the zero-measure clause is a separate open question
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number-theory Jun 24, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → disproved
For S ( x ) = # { ( a , b ) : a + b ≤ x , σ ( a ) + σ ( b ) = σ ( a + b ) } S(x) = \#\{(a,b) : a + b \le x,\ \sigma(a) + \sigma(b) = \sigma(a+b)\} S ( x ) = # {( a , b ) : a + b ≤ x , σ ( a ) + σ ( b ) = σ ( a + b )} , is S ( x ) ∼ c x S(x) \sim cx S ( x ) ∼ c x ? The preprint claims S ( x ) S(x) S ( x ) grows faster than x ( log x ) R x (\log x)^R x ( log x ) R for every fixed R R R , ruling out the linear asymptotic.
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combinatorics Jul 22, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
Improves the exponent rather than answering the asked question: whether an N o ( 1 ) N^{o(1)} N o ( 1 ) colouring exists remains open.
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geometry-topology Jul 13, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
superlinear lower bound M(n) ≥ n^{1 + 1/(50000 log log n)}, improving Ω(n log n); the exact order remains open
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number-theory Jan 17, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
VibeMathed records this result as “Erdős Problem #281.” The registry entry and named primary source contain the available statement and scope.
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combinatorics Jul 26, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
closes a one-block gap in the covering tables; the analogous next case is not reachable by this method
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quantum-information-computing May 21, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → disproved
the diagonal family is ruled out; the broader two-parameter problem remains open
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geometry-topology Jul 20, 2025 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
A paper torus is an embedded polyhedral torus isometric to a flat torus. Schwartz proves no paper torus with 7 vertices exists and constructs one with 8, settling the minimum-vertex question in the flat-torus embedding tradition of Császár-torus combinatorics and the Lazarus-Tallerie universal triangulation.
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number-theory Jan 10, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
VibeMathed records this result as “Erdős Problem #729.” The registry entry and named primary source contain the available statement and scope.
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number-theory Apr 19, 2026 Significance 10/100 Registry: site confirmed
Prior state unknown → proved
VibeMathed records this result as “Erdős Problem #1195.” The registry entry and named primary source contain the available statement and scope.
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algebra Aug 8, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
The footnote is worth reading on its own: an author saying in print that the model earned coauthorship and that policy is what prevents it.
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analysis Apr 30, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
An elementary solution via a primitive-row decomposition of the Chebyshev-node measures; the main theorem is formalized in Lean, but erdosproblems.com still lists the problem open
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combinatorics Aug 10, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
The key step is a lower bound on the truncated imbalance sum, which yields every Erdos-Gallai inequality for the sorted imbalance list; a parity computation finishes it.
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combinatorics Jul 13, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
exact k = 3 constant established, settling the Parrilo-Robertson-Saracino conjecture for 3-APs; general k remains open
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number-theory Apr 27, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
VibeMathed records this result as “Erdős Problem #42.” The registry entry and named primary source contain the available statement and scope.
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number-theory Jun 14, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
lower bound improved to ≫ log n/(log log n · log log log n) infinitely often; the extremal order remains open
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theoretical-computer-science Jul 23, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
Two open problems about extracting order from trees in real-valued functions. A quantitative function analogue of Hodges's tree-to-order extraction yields an at most double-exponential bound on dual sequential fat-shattering dimension, resolving the first. A new proof of Daskalakis-Golowich tight-threshold extraction, avoiding multicolored Ramsey numbers, resolves the second, which concerned repairing the bound in a…
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number-theory Jul 13, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
For the least t k ( n ) t_k(n) t k ( n ) with n ∣ t k ( n ) ( t k ( n ) + 1 ) ⋯ ( t k ( n ) + k − 1 ) n \mid t_k(n)(t_k(n)+1)\cdots(t_k(n)+k-1) n ∣ t k ( n ) ( t k ( n ) + 1 ) ⋯ ( t k ( n ) + k − 1 ) , do the conjectured logarithmic-saving and adjacent-length estimates hold on average? Both answered affirmatively, with c = 1 / 2048 c = 1/2048 c = 1/2048 admissible in the t 2 t_2 t 2 bound.
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number-theory Mar 31, 2026 Significance 10/100 Registry: site confirmed
Prior state unknown → proved
VibeMathed records this result as “Erdős Problem #380.” The registry entry and named primary source contain the available statement and scope.
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combinatorics Jul 29, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → disproved
Around the Kemeny median problem, which stays open for m = 3 m=3 m = 3 and m = 5 m=5 m = 5 voters, the paper refutes three conjectures on tournament inducibility: both conjectures of Milosz, Hamel and Pierrot (the 3-cycle extension for odd m ≥ 5 m\ge5 m ≥ 5 , and F A S = H S 3 \mathrm{FAS}=\mathrm{HS}_3 FAS = HS 3 at n = 11 n=11 n = 11 ), and Shepard's threshold conjecture.
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number-theory Dec 26, 2025 Significance 10/100 Registry: lean verified
Prior state unknown → disproved
VibeMathed records this result as “Erdős Problem #897.” The registry entry and named primary source contain the available statement and scope.
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number-theory Jul 1, 2026 Significance 10/100 Registry: site confirmed
Prior state unknown → proved
Let S ( N ) S(N) S ( N ) count the distinct values of ∑ n ∈ A 1 / n \sum_{n\in A} 1/n ∑ n ∈ A 1/ n over A ⊆ { 1 , … , N } A\subseteq\{1,\dots,N\} A ⊆ { 1 , … , N } . Estimate S ( N ) S(N) S ( N ) .
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number-theory Apr 6, 2026 Significance 10/100 Registry: site confirmed
Prior state unknown → disproved
The question as posed was implicit in Davenport–Erdős (1951); the AI result settles Tenenbaum's open variant negatively
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number-theory Apr 30, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
As Tao notes on the problem page, the claim establishes natural LOWER density at least 1-eta but not that the natural density exists, so the problem as stated remains technically open
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