Borsuk's Conjecture Fails in Dimension 9, Sidorenko's Conjecture and 8 More

Constructs a compact subset of ℝ9 that cannot be covered by ten sets of strictly smaller diameter, disproving Borsuk's covering assertion already in dimension nine. The example consists of rank-one orthogonal projectors onto lines in ℝ4, with the Frobenius metric. (Lean)

Disproves Sidorenko's conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not force quasirandomness. (Lean)

Disproves Ryser's covering conjecture by constructing intersecting (q+1)-partite, (q+1)-uniform hypergraphs with covering number q+1, rather than the predicted bound q, for every sufficiently large prime q. A separate construction over extension fields also disproves Gyárfás's monochromatic tree-cover conjecture. (Lean)

Proves Hindman's finite sums and products conjecture: every finite coloring of the positive integers contains sets of any prescribed finite size whose nonempty subset sums and nonempty subset products all have one common color.

Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but o(n) points of any n-point planar set determine at least n^1-ε distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by O(n^4/3-δ) for an absolute δ > 0. (Lean)

Resolves the full combinatorial invariance conjecture: isomorphic Bruhat intervals in arbitrary Coxeter systems have identical equal-parameter Kazhdan–Lusztig polynomials. Thus the abstract order of the interval determines the polynomial, even across different Coxeter systems. (Lean)

Resolves the elementary-positivity part of the Shareshian–Wachs conjecture: the chromatic quasisymmetric function of every natural unit interval graph has elementary-basis coefficients in ℕ[q]. The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity. (Lean)

Proves Seymour's second-neighborhood conjecture: every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one. Oriented graphs may be arbitrary apart from the exclusion of loops and oppositely directed edge pairs. (Lean)

Proves that integral and fractional expectation thresholds differ by at most a universal factor, and resolves Talagrand's discrete-convexity conjecture. An application proves the Ascoli–He–Park–Talagrand graph-decomposition conjecture: every graph's edges split into a universally bounded number of fixed pieces, each with containment threshold at most a universal constant times the original graph's integral expectation threshold. The pieces' embeddings need not agree on shared vertices. (Lean)

Proves the second Kahn–Kalai conjecture: for every finite simple graph H with h ≥ 1 edges and at most n vertices, its appearance threshold in G(n,p) is at most C p_E(n,H)(1+log₂ h), with universal C. Here p_E is the least density at which every subgraph of H has expected copy count at least 1/2. (Lean)