The Ramsey number $R(m,n)$ is the smallest order at which every red-blue edge coloring of a complete graph must contain a blue clique (a complete subgraph) of size $m$ or a red clique of size $n$. Determining these numbers exactly is extremely hard, and even certifying a lower bound requires exhibiting an explicit coloring that avoids both cliques. We develop an integer programming framework for certifying such lower bounds, restricting the search to circulant graphs, whose rotational symmetry lets us reformulate the problem in a projected distance space, reducing the number of binary variables from quadratic to linear in the graph order. We strengthen this projected model through coefficient reduction and solve it with a branch-and-cut algorithm whose separation routine exploits the common neighborhood structure of circulant graphs, combining heuristic and exact maximum-clique algorithms. In an extensive computational campaign on circulant graphs with up to 410 vertices, we improve the best lower bounds previously obtained by other methods by up to 11 points for 25 values of $R(3,n)$ with $24\le n\le49$ and $n\neq27$, each backed by an explicit graph certificate that can be independently verified with a stand-alone exact clique solver. To the best of our knowledge, our method also provides the first reproducible optimization-based procedure for certifying circulant Ramsey numbers $R_C(m,n)$, which we use to establish eight new values of $R_C(3,n)$ with $13\le n\le20$. Our framework, graph certificates, and stand-alone checker are provided as supplementary material to support independent verification and reuse.
For integers $k,g \ge 3$ let $n_g(k)$ denote the minimum order of a graph with chromatic number $k$ and girth at least $g$. Exoo and Goedgebeur (DMTCS 2019) proved $26 \le n_6(4) \le 66$; their 66-vertex witness has remained the smallest known 4-chromatic graph of girth 6. We improve both bounds to $29 \le n_6(4) \le 64$. The upper bound is witnessed by an explicit 4-chromatic graph of girth 6 on 64 vertices with 152 edges; it is vertex- and edge-critical, and its automorphism group is cyclic of order 8 and acts semiregularly. The lower bound is an exhaustive isomorph-free computation in the SAT modulo symmetries framework with co-certificate learning, driven by the Liu-Postle edge-density bound for 4-critical graphs of girth five; it re-derives $n_6(4) \ge 26$ by a disjoint method and is validated on the known values $n_4(4)=11$ and $n_5(4)=21$. We complement the bounds with structural obstructions: no smaller witness arises from either known witness by local modifications; no 4-chromatic Cayley graph of girth 6 exists on 54-63 vertices (for orders 59 and 61 no vertex-transitive witness exists at all); and no witness on at most 63 vertices admits a semiregular automorphism group with two or three vertex orbits, for any finite group. Since every known witness of an $n_g(4)$ record with $g \ge 6$ is a lift of a small base graph along a semiregular action, these results close the most symmetric part of that regime below 64 vertices. All properties of the new graph are verified by independent programs and formally certified in the Lean 4 proof assistant: the non-3-colourability is established inside Lean by a formally verified checker that re-validates a 219,532-node refutation certificate, with a machine-checked soundness theorem.
A {\it threshold graph} is a graph that can be constructed from the one-vertex graph by repeatedly adding either a dominating vertex or an isolated vertex. Motivated by an induced Ramsey-type problem for this class, we define $r'_2(s)$ to be the minimum integer $n$ such that every $n$-vertex graph contains an induced threshold graph on $s$ vertices. We establish exponential upper and lower bounds for $r'_2(s)$ and determine its exact values for $s\in\{3,4,5,6\}$. To study this problem from an edge-coloring perspective, we use the notion of an orderable coloring, introduced by Richer [{\it J. Combin. Theory Ser. B}, 80(1) (2000), 172--177]. An edge-colored graph is {\it orderable} if its vertices can be ordered so that, for each vertex, all edges from it to later vertices have the same color. Equivalently, $r'_2(s)$ is the minimum $n$ such that every $2$-edge-coloring of $K_n$ contains an orderable $K_s$. We also determine the exact value of the unordered canonical Ramsey number $CR(s, 3)$ for all $s \ge 3$, where $CR(s,3)$ denotes the minimum integer $n$ such that every edge-coloring of $K_n$ contains either an orderable $K_s$ or a rainbow $K_3$. More generally, for graphs $G$ and $H$, we study $r'_2(G)$, the corresponding $2$-color Ramsey number for an orderable $G$, and $CR(G,H)$, where the alternative is a rainbow $H$. For complete bipartite graphs, we prove that for every fixed $s$, $r'_2(K_{s,t}) = CR(K_{s,t}, K_3)= \left(\frac{2^s}{s+1}+o(1)\right)t$ as $t\to\infty$. For $s\in \{2,3\}$, we further determine the exact values of these parameters for infinitely many $t$, using constructions arising from strongly regular graphs, Hadamard matrices and conference matrices.
The simplex of complete graphs, also known as the first-order truncated simplex lattice, is a network of $M+1$ identical complete graphs, each with $M$ vertices, such that each clique contains an edge or bridge to every other clique. It contains $N = M(M+1)$ vertices, and previous asymptotic results using a continuous-time quantum walk to search this graph for a single marked vertex have either numerically demonstrated an optimal runtime of $O(\sqrt{N})$, or analytically proved a deterministic success probability of 1, but not both, even when the bridges are weighted. In this paper, we give the first analytical proof of optimal quantum search on this graph, proving that it occurs when the weight of the bridges equals $M$. In addition, we numerically show that the optimal runtime is achieved more broadly whenever the weight is at least $\sqrt{M}$. Furthermore, the algorithm is also deterministic when the weight scales between $\sqrt{M}$ and $M$, and this is the first example of quantum search on the simplex of complete graphs that is both asymptotically optimal and deterministic. In addition, for weights where the algorithm is nondeterministic, we give a way to find the marked vertex by inspecting neighboring vertices. Finally, while it is known that connectivity is not a reliable indicator of fast quantum search when comparing different graph families, we show that it is also unreliable within the graph family of weighted simplex of complete graphs.
Kiyoji Huang Fujiwara, Yujia Shi, Thomas G. Wong· 0 citations
Recently, Chudnovsky, Cook, Davies, Oum, and Tan obtained the first finite bound on the chromatic number of t-perfect graphs, showing that they are 199053-colorable. We improve this bound to 186 by refining their proof. The original proof establishes that every graph with large odd girth and large chromatic number contains a certain structure called an r-arithmetic rope, and that its existence in a certain leveling of a graph with large odd girth would imply an odd wheel as a t-minor, a known obstruction of t-perfectness. While their technique requires a lower bound on the chromatic number that is exponential in r, we show that the existence of an r-arithmetic rope can already be guaranteed under a linear bound. Using a slightly weakened notion of arithmetic ropes allows us to reduce the bound even further.
The Peterson hit problem seeks a minimal set of generators for the polynomial algebra $\mathcal P_k=\mathbb F_2[x_1,\ldots,x_k]$ as a module over the mod--2 Steenrod algebra. While completely resolved for $k \leq 4$, the unrestricted problem remains widely open for $k \geq 5$, where the combinatorial explosion of basis elements renders exact algorithmic computation intractable. To bypass full Gaussian elimination, we model the degree--$d$ hit space via a sparse matrix driven by the Cartan formula and Lucas's theorem, shifting the focus to the construction of certified matrix minors. We first prove that strict spike monomials exactly characterize the zero rows, establishing a hard structural limit on coordinate-level annihilators. To bound the matrix rank from above (cohit lower bound), we derive exact zero-column formulae, which are strictly refined by the exact homology of the $\operatorname{Sq}^1$-layer and systematic linear dependencies induced by Adem relations. To bound the rank from below (cohit upper bound), we extract explicit independent column families: singleton columns yield permutation minors, acyclic pivot systems optimize triangular minors across all row orders, and $q$-support columns are formalized through hypergraph incidence. Crucially, we identify a congruence family that decomposes precisely into simplicial boundary matrices over $\mathbb F_2$, yielding a sharp closed-form rank formula. The resulting two-sided bounds are universally computable for every $k \geq 1$ and $d \geq 0$. Significantly, these results establish the absolute limits of purely combinatorial approaches to the hit problem, cleanly separating universal discrete certificates from the degree-specific resolutions provided by representation theory and weight filtrations.
Gupta and Iosevich introduced the edge complexity of a graph as the minimum Fourier ratio of its adjacency matrix over all vertex labelings and bounded it below by graph energy divided by the square root of twice the number of edges. We characterize equality for a fixed labeling: the Fourier transform of the adjacency matrix must have at most one nonzero entry in each row and column. This implies regularity, circulancy of every positive even power of an extremizing adjacency matrix, and a parity restriction on connected components, and it gives equality results for certain Laplacian spectral projectors. We construct equality cases from affine involutions on cyclic groups. Singer difference sets yield, for every prime power $q$, an equality-attaining $(q+1)$-regular graph that is not an abelian Cayley graph. We also establish Fourier-ratio estimates for weak, Cartesian, and strong graph products, including preservation of equality under weak products of coprime orders. We use Fourier-ratio recovery as a coding theorem to obtain entropy upper bounds for low-complexity adjacency matrices and complement them with a lower bound obtained by perturbing complete graphs. Finally, a concentration argument shows that if $Np_N/\log N\to\infty$ and $\limsup_{N\to\infty}p_N<1$, then $\operatorname{FR}_{\min}(G(N,p_N))$ is of order $N$ with probability tending to one.
Vishal Gupta, A. Iosevich, J. Iosevich et al.· 0 citations
Known for his clear and elegant writing style, Bertsekas shaped fields from control and optimization to large-scale computation and artificial intelligence.
