Problems

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…

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Han and Xiong extended the Gaussian binomial coefficient to positive rational index and conjectured that its integer trace, the integer-exponent part of the resulting power series, is coefficientwise largest at the integer point. Ono's…

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Erdős asked whether every n-point set in Euclidean space whose pairwise distances are mutually at least 1 apart must have diameter at least (1+o(1))n^2. Disproved: an explicit high-dimensional construction beats the conjectured constant.

If W(k) is the least N such that every two-colouring of 1, …, N contains a monochromatic k-term arithmetic progression, must W(k+1) - W(k) → ∞?

For a single-source unsplittable flow, find the optimal universal additive constant C s.t. every feasible fractional flow x with arc costs c should admit an unsplittable routing y with c^top y ≤ c^top x and y_a ≤ x_a + C · D on every arc.…

Can the edges of a finite connected multigraph, given a closed eulerian trail, be partitioned into circuits so that no circuit contains two edges used consecutively in the trail? The proof in fact four-colours the edges to satisfy the…

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…

Let N(k, ℓ) be the least N such that every f : [N] → -1, 1 has a k-term arithmetic progression P with |∑_n ∈ P f(n)| ≥ ℓ. In particular, is N(k, 2) ≤ C^k?

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…

Donner proved in 1992 that the list color function P_ℓ(G,k) equals the chromatic polynomial P(G,k) once k is large. Kaul and Mudrock asked whether the analogue holds for Hanlon's unlabeled chromatic polynomial, and could not settle even…

Let f_3(N) be the least size forcing a set A ⊆ 1,…,N to contain distinct a,b,c with a+b, a+c and b+c all in A. The upper bound f_3(N) ≤ 5N/8 + O(1) matches the standard construction [N/8,N/4] ∪ [N/2,N], so f_3(N) = 5N/8 + O(1).

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

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.

Ehrhart equivalence is a necessary and sufficient condition for (not necessarily finite or rational) discrete equidecomposability.

Is it true that for every nonnegative integer k, there exists a connected graph G satisfying φ(G) − κ(G) + 1 = k?

Is there a nice combinatorial proof for the number of interior lattice points of P_n(132,312) ?

We discuss ... including some new ones that we present in this last section (in particular Conjectures 11.3 and 11.5).

Can Theorem 4.2 be true for dimension ≥ 4 ?