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For a bipartite graph without isolated vertices and with a unique minimum dominating set, does the proposed function bound the number of edges whenever and ? A -vertex bipartite graph with edges exceeds the conjectured maximum of .
For every , is the independence polynomial of real-rooted if and only if is even? Exact Sturm counts refute both directions.
The formalization proves the statement under the weaker hypothesis n >= 2; the pull request marking the conjecture solved is open, not merged
For every connected graph , is , where is the largest induced-bipartite-subgraph order? An -vertex counterexample - a triangle with four leaves on each of two vertices - has against bound .
conditional on a plausible number-theoretic conjecture; unconditional through s = 15
For , can the full arc set be partitioned into three directed Hamilton cycles for every integer ?
This settles Steinerberger's third open question and separates the two models: in the disk, full chords cost you a factor growing like over what is achievable without the restriction. It also improves the Steinhaus-type to polylogarithmic within the full-chord class. It does not settle the Buffon discrepancy problem itself. Steinerberger's first question - whether every convex body admits a se…
For an odd prime , do Sun's normalized trigonometric permanents satisfy and ? Exact computation at refutes both sign laws.
In the all-heads coin game a player starts with coins, each showing heads with probability ; each round all remaining coins are flipped, the player must set aside at least one head (losing if none shows), and wins once all coins are set aside. Determine optimal strategies and the winning probability . Resolved: for every strategy achieves ; for the single…
For every finite connected graph, is at most the product of its largest induced-tree order and its second-smallest degree?
If is connected, cubic and diamond-free, must the zero-forcing number satisfy ? A connected cubic triangle-free -vertex graph has and .
The a = 3 slice is settled outright. The parent conjecture's dihedral side has since been resolved for every a >= 4 as well (see the related entry), so Conjecture 4.9's claim 1 + (a-1)(b-1) now stands proved for all a >= 3; the trivial a = 1, 2 cases and the cyclic analogue for a >= 4 remain formally unaddressed.
For the Fubini numbers , is for every , as conjectured on the OEIS in 2018?
Is the zero forcing number of every connected graph with maximum degree at most its independence number plus one? A connected 24-vertex subcubic graph with independence number and zero forcing number refutes this 2017 TxGraffiti conjecture, and a 36-vertex cubic variant refutes the cubic form: is attained.
refuted under the standard-deviation reading; the statement is reading-sensitive and other readings remain open
WOW-284 asserts that the minimum dual degree of every connected graph of order at least three and girth at least five is at most the negative of its least distance eigenvalue. The paper refutes it with exact counterexamples of orders 38, 39, 40, 42 and 50, and develops a structural theory of the failure.
Does a single ReLU neuron trained on modular addition align to one Fourier frequency? Open Problem MAIS-O60, itself posed by Claude Fable 5 under the direction of Lionel Levine, is answered negatively: an explicit construction reaches a frozen state whose Fourier energy is spread equally across all nonzero frequency classes, on an open set of initial conditions.
Answered in full: for every , a target with exactly two nonadjacent zero digits has , independently of the distance between the zeros. Exact at every width, no error term, no hypothesis on (Theorem 1.1). This is an evaluation, not an extremal result, and the paper is explicit about the difference: the plateau value is not maximal. At it reads while…
Sivaraman asked whether perfect divisibility is characterized by its chromatic consequence: is a graph perfectly divisible if and only if for every induced subgraph of ? False: the Paley graph satisfies the chromatic bound hereditarily but is not perfectly divisible.
Fixed four-terminal planar DAG; positive route-cost differences for m≥2, with non-attained suprema; signed/zero only for m=1, whose attainment is unstated. No topology-wide/unrestricted-planar claim.
Agulnick and Busick-Warner exhibited a family of fifth-order ambiguities on the exact unit support and conjectured that it was not a complete classification because it did not use the full field . This work proves the complete classification. The larger field enlarges the common amplitude , while every relative ambiguity remains a norm-one parameter in . Th…
Nothing in the paper's proof is affected. At the witness, inequality (4) holds with slack +27.0, inequality (5) holds with equality (f is inner), and the theorem itself holds with slack 2 - kappa = +0.80. Only the shortcut Remark 2 floats is refuted.
no constant-bound repair of the conjecture is possible
Under smoothness, positivity, decay and score assumptions, are all steady solutions of the Coulomb Vlasov-Maxwell-Landau system on necessarily spatially uniform Maxwellians?
Opus 4.8 constructed a branch of the stated depth, giving a lower bound, and believed it had a matching upper bound; that proof was wrong and the statement stayed a conjecture. FABLE 5 later proved it. In the author's summary of the method: "The proof turns conjugation, near , into base- arithmetic." A cycle near the fixed point splits into a coarse base permutation and a vector of carries in ,…
A tournament orients every pair in a round-robin (winner → loser). The score sequence is the sorted win-count list. Reversing a directed 3-cycle never changes scores, so score-equivalent tournaments can look structurally different. Question: Which linear combinations of induced k-subtournament type-counts are score-determined — identical across all tournaments sharing a score sequence, at any host size? Answer: Ex…
Finite-jet theorems about compiled truncated EMD equation certificates: the exact shear-orbit fiber classification of the complete first seed channels, an explicit collision family with one metric three-jet realized by an actual cubic metric germ (genuine Frechet Ricci value and first derivative), the compiled impossibility theorem, and the fourth-order recovery with equality fiber . Not settled here: promo…