Problems
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For a graph G, its vertex deck is the multiset of graphs obtained by deleting one vertex. Bowler, Brown, and Fenner (BBF) proposed 2⌊(n−1)/3⌋ as the maximum possible overlap between the decks of two nonisomorphic n-vertex graphs, for all…
Albertson and Berman conjectured that for every simple planar graph G on n vertices, the largest vertex set inducing a forest has size at least n/2. The standing lower bound since the same year has been Borodin's 2n/5, from his acyclic…
How dense can a sum-free subset of the lattice cube 1,…,n^d be? Aydinian and Cameron asked for the limiting density, which is also Problem 6 in Ben Green's list of 100 open problems. The natural conjecture is that the optimum is a slice x…
In the Frankl-Pach-Erdős circle of VC-dimension problems, the first arXiv version of the paper posed the k=3 case of a witness construction question. ChatGPT 5.4 Pro answered it; the published construction generalizes the model's response,…
Deng, Tidor and Zhao asked whether [N] admits a coloring with N^o(1) colors and no symmetrically coloured 4-term arithmetic progression, giving an O(N^log_223) coloring. The paper gives an O_k(N^4/k^2) coloring of [N] avoiding…
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…
Reiner conjectured a description of the homotopy types of intervals in higher Bruhat orders. In corank 3 it holds: the facial intervals of B(n,n-3) are exactly the spherical intervals, and every other interval is contractible.
Determine the Shannon capacities of odd cycles beyond C_5, or improve the best explicit bounds. Lovasz's theta function settled C_5 in 1979 and every longer odd cycle has stayed open since. The current records, all obtained with model…
Is the number of nonnesting permutations of 1,1,…,n,n avoiding both 1132 and 3312 equal to 3^n - 3 · 2^n-1 + 1 for every n ≥ 1?
VibeMathed records no statement for this problem. See formal-conjectures PR #4668 - Mark WOWII Graph Conjecture 217 solved for the original.
Jaeger conjectured that every bridgeless cubic graph G admits a Petersen coloring: a map φcolon E(G)→ E(P) into the edges of the Petersen graph P such that, for every vertex v of G, the three edges at v are sent to three edges meeting at a…
Simonovits conjectured that if a forbidden family F with p(F) > 1 has extremal number exceeding the Turan bound by a superlinear surplus, then its extremal graphs are joins of p graphs, each extremal for a family of chromatic number two.…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Let A(k) be the largest possible number of moves in a north-east lattice path whose visited vertices contain no k collinear points. Gerver (1979) and Gerver and Ramsey (1979) bounded A(k) by exp(Ω(log(k)^2)) ≤ A(k) ≤ exp(O(k^4)), and…
Korsky, Saffat and Aiylam bounded the growth constant c(G) for integer-valued Lipschitz functions on G(n,d/n) between 1/(2d) and 4log^2 d/d up to lower-order terms. The random-graph side is sharpened.
A graph on n vertices is very well-covered if every maximal independent set has size n/2. Levit and Mandrescu conjectured that the independence polynomial i(G,x) of every very well-covered graph is unimodal, i.e. its coefficient sequence…
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.
An asymptotic formula for p(k), the limiting probability that a random permutation has an invariant set of size k: it is asymptotically k^-δ(1+o(1)) times a smooth positive function, sharpening a line of estimates running through…
The multivariate independence polynomial is the partition function of the hard-core model with per-vertex fugacities. The paper proves a lower bound extending to the multivariate setting a result Tao proved in the univariate case, and…
Teschner conjectured that every finite simple graph G with at least one edge satisfies b(G) ≤ 3/2Δ(G), where b(G) is the bondage number and Δ(G) is the maximum degree. Yavari gives a connected cubic bipartite graph on 18 vertices with…