We present a complete computer-assisted verification of the Antichain Width Conjecture for all finite partially ordered sets (posets) of width at most 6. The conjecture asserts that in any finite poset of width w, the maximum antichain can be partitioned into at most w chains that collectively cover the antichain.
The minimum dominating set problem in Kneser graphs K(n,k) is a classical question in combinatorial optimization, yet the monotonicity of the domination number gamma(K(n,k)) in n for fixed k has remained unresolved for k >= 3. We introduce the Spectral Degeneracy Index (SDI), defined as the ratio of the second-largest eigenvalue to the algebraic connectivity, and prove that non-monotonicity of gamma occurs precisely when SDI exceeds an explicitly computable threshold tau_k.
Compute Ramsey numbers R(K_{s,t}, K_{s,t}) for s,t ≤ 8 via SAT solving and ILP. The growth rate exhibits a sharp transition: for t/s < 0.
Identifying codes, introduced by Karpovsky–Chakrabarty–Levitin, are useful for fault localization in networks. In the binary Hamming space (hypercube) Q_n, let M_r(n) denote the minimum size of an r-identifying code.