Problems

For a connected graph G , let t= tree( G ) (order of a largest induced tree), A= average eccentricity, and L= maximum independence number of a neighbourhood. Then ⌈ (A+L)/3 ⌉ ≤ t. (The evenly-divided reading of the conjecture holds; a…

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.