Problems

For every finite connected simple graph G, is the order of the largest induced tree at least girth(G) - 1 + ecc(G, center(G)), where the last term is the eccentricity of the centre set? Answered affirmatively, with a Lean proof.

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.

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).

Let X be a set of cardinality ℵ_ω and f a function from the finite subsets of X to X such that f(A)not∈ A for all A. Must there exist an infinite independent Y⊆ X, i.e. with f(B)not∈ Y for all finite B⊂ Y? Claimed resolution: the positive…

Does every nontrivial finite simple graph have noninteger Sombor energy? If ρ_1,…,ρ_n are the eigenvalues of the Sombor matrix of a graph G, its Sombor energy is E_SO(G)=∑_i=1^n|ρ_i|. The conjecture asserted that E_SO(G)∉ Z for every…

Let R(3;k) be the least n such that every k-colouring of the edges of K_n contains a monochromatic triangle. Determine lim_k→∞ R(3;k)^1/k (a $250 Erdős prize problem). A superexponential lower bound resolves the problem: the limit is…

Let G be a simple connected graph on n≥ 5 vertices. If the maximum over all vertices v of ℓ(v) - the independence number of the subgraph induced by the open neighborhood N(v) - is at most 1, must G be well totally dominated? Answered…

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…

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

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.

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…

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 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.