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If each integer has at most r representations m = pa with p prime and a ∈ A ⊆ [1, N], what is the best upper bound for ∑_a ∈ A 1/a? The candidate proof gives the matching order Θ_r(log N / loglog N).

VibeMathed records no statement for this problem. See formal-conjectures PR #4668 - Mark WOWII Graph Conjecture 217 solved for the original.

Let F(N) be the maximal size of A⊆1,…,N such that no a∈ A divides the sum of any nonempty subset of A∖a. Estimate F(N). The lower bound F(N)≫ N^1/5 is classical, from constructions of Erdős and Csaba, and every non-dividing set is…

If H is bipartite and r-degenerate, is ex(n;H) ≪ n^2-1/r (a $500 Erdős-Simonovits prize conjecture)? A counterexample refutes the degeneracy conjecture.

If a/b ∈ Q_>0 and b is squarefree, can a/b always be written as a finite sum of reciprocals of distinct products of two distinct primes?

The dimension-five case asks whether, for every nonnegative 5×5 real matrix A whose entries sum to 5, the Dittert functional Φ(A)=∏_i r_i+∏_j c_j-per(A) is uniquely maximized at U_5=J_5/5. The submitted artifact claims the stronger…

Let a_1 = 2 and a_2 = 3 and continue the sequence by appending to a_1, …, a_n all possible values of a_ia_j - 1 with i ≠ j. Is it true that the set of integers which eventually appear has positive density?

If A ⊆ N has unbounded dyadic-shell counts and ∑_n ∈ A |θ n| = ∞ for every 0 < θ < 1, must A be complete - is every sufficiently large integer a sum of distinct elements of A?

Which finite triple systems occur in every triple system of uncountable chromatic number? The claimed characterization: exactly those that, after removing isolated vertices, are linear, have every hyperedge-node of their Levi graph meeting…

For a finite forbidden triple system G, what exact uncountable chromatic cardinalities occur among G-free triple systems, and how do those spectra interact? The revised manuscript answers the three exact-cardinal questions and claims a…

For the least t_k(n) with 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 admissible in the t_2 bound.

For every finite family F of graphs, is there a single G ∈ F with ex(n;G) ≪_F ex(n;F)? A counterexample refutes the Erdős-Simonovits compactness conjecture.

Determine the leading asymptotic of the largest eigenvalue of the N-Majorana quartic SYK Hamiltonian as N → ∞. The preprint proves λ_1/√N → 4∫_0^∞ g_0(t)^4 dt ≈ 0.32504 almost surely, via the limiting free energy at every fixed positive…

Does there exist an integer polynomial f of degree at least two and a set A ⊆ Z such that every integer has a unique representation n = a + f(k)? A manuscript claims the thirteenth powers admit a tiling complement.

Let m_1≤…≤ m_k and n be sufficiently large. If T is a tree on n vertices and G is the complete multipartite graph with vertex class sizes m_1,…,m_k, prove that R(T,G)≤ (χ(G)-1)(R(T,K_m_1,m_2)-1)+m_1.

Question 8 of the First Proof experiment (Abouzaid et al.) asks whether a polyhedral Lagrangian surface with exactly four faces meeting at every vertex necessarily admits a Lagrangian smoothing. The research report assembles…

Every finite connected simple graph G satisfies α(G)≥ r(G)+ln(ρ(G)), where α(G) is the independence number, r(G) is the radius, and ρ(G) is the minimum number of pairwise vertex-disjoint paths whose vertices cover V(G).

Among all nonconstant monic polynomials f whose roots lie in [-1, 1], determine inf_f |x ∈ R : |f(x)| < 1|.

Given a binary matrix M, decide whether its columns can be permuted so that every row contains at most two blocks of 1s and, if it contains two blocks, they are separated by at most one 0. The claimed theorem proves that this (2,1)-Gapped…

For P_n(z) = ∑_k=0^n ε_k z^k with independent uniform signs, does the number R_n of roots in |z| ≤ 1 satisfy R_n/(n/2) → 1 almost surely? The manuscript proves the strong law with R_n = n/2 + O_ω(n^149/150).