Papers

Let $G_n$ be the coprime graph on $\{1,\ldots,n\}$. We prove that the mixed vertex-coloring coprime Ramsey number satisfies \[ \Rcop(k_1,\ldots,k_c)=p_{\sum_{i=1}^c(k_i-1)}, \] where $p_m$ is the $m$-th prime. The proof is elementary: the prime clique $\{1\}\cup\{p\le n:p\text{ prime}\}$ gives the upper bound by pigeonhole, while a prime-bin partition gives the matching lower bound by coloring each composite with a bin containing one of its prime divisors. We reserve $\Rcop$ for this vertex-coloring parameter; the edge-coloring parameter on the same host graph is denoted $\Redge$. The same certificate viewpoint yields several extensions, including a support-disjointness generalization, a polynomial-time certificate-extraction primitive, and an exact reduction of the edge-coloring variant to classical Ramsey numbers: $\Redge(k_1,\ldots,k_c)=p_{\Rcl(k_1,\ldots,k_c)-1}$. These two formulas are rank transfers from the same clique-label certificate. We also prove that the balanced two-color diagonal threshold equals the unrestricted threshold $p_{2k-2}$ for all $k\ge2$, via a deterministic prime-bin split requiring only the weak inequality $2p_m<p_{2m}<3p_m$; for fixed $c$, a Hall argument plus a standard Selberg–Delange estimate gives eventual multicolor balanced certificates.

We study an opinion dynamics model in which $n$ agents hold directed trust or distrust opinions about one another, represented as a matrix $M \in \{0,1\}^{n \times n}$ in which 1 represents trust and 0 represents distrust. A gossip event $(a, z, y)$ causes agent $z$ to adopt agent $a$'s opinion of $y$, provided that $z$ trusts $a$. We characterize the absorbing states of this process, i.e. the states in which no further opinion change can take place: we find that they are the states in which agents are partitioned into isolated factions, each faction containing a subset of core members who share mutual trust, while the remaining peripheral members trust all core members but receive no trust in return. This structure establishes a bijection between absorbing states on $[n]$ and pairs consisting of a set partition $\pi$ of $[n]$ together with a choice of non-empty subset of each faction of $\pi$. The number of such absorbing states is therefore given by OEIS A143405, with exponential generating function $\exp(\exp(x) \cdot (\exp(x) - 1))$. In addition, up to isomorphism, the count equals the number of plane partitions of $n$, given by OEIS A000219, recovering MacMahon's classical product formula $\prod_{k \geq 1} 1/(1 - x^k)^k$. Exhaustive computation for $n \leq 7$ confirms both counts.

We generalize and prove the recursive relation \[ S_{\lambda}(V) = \sum_{L\subseteq V\text{ line}} S_{\lambda}(V \mathbin{/\mkern-5mu/} L) \] conjectured by I. G. Macdonald for his "7th variation" of the Schur functions. This variation is a family of polynomials over a finite field that mimic the (straight and skew) Schur polynomials using powers of the Frobenius.

We give an explicit strong deformation retraction of the complement of a Fermat hypersurface of degree $d$ in the complex projective space of dimension $n$ into a $\Delta$-complex of real dimension~$n$. This retraction is performed in several steps, using the branched cover structure of the Fermat hypersurfaces to the degree $1$ case. Moreover, an implementation of the method in Sagemath is provided.

This paper establishes new upper bounds for the sum of the $k$ largest eigenvalues of symmetric matrices. When applied to the adjacency matrix of a graph, our results improve upon a related bound due to Mohar [On the sum of k largest eigenvalues of graphs and symmetric matrices, J. Combin. Theory Ser. B 99 (2009) 306–313]. Furthermore, in the case of the Laplacian matrix, we prove that the well-known Brouwer's conjecture [Spectra of Graphs, Springer, New York, 2012] holds for small values of $k$ for almost all graphs, thereby taking a significant step toward its complete resolution.

We propose a degree-preserving variant of the Builder–Chooser clique game of Pettie, Tardos, and Walczak. In each round, Builder chooses a matching, performs a degree-preserving growth (DPG) step by replacing the chosen edges with edges incident to a new vertex. Then partitions the entire edge set into two parts, and Chooser keeps one part. We begin the study of this game with the first nontrivial target, namely forcing a triangle. For triangle-free initial graphs we prove an exact one-round criterion, derive an exact one-round threshold on paths and exact forcing times on cycles, and identify the $5$-cycle as the first genuine two-round example. We then formulate a one-round criterion for larger cliques, prove a sharp exact result for forcing $K_4$ from triangle-free seeds. We establish general lower bounds on clique-forcing times from clique-free seeds, and isolate a conjectural template-amplifier lemma which, if proved, would imply that every clique is forceable from some triangle-free seed.

This paper proves a conjecture by Solomon about Steiner shallow-light trees (SLT) in Euclidean $d$-space: It is shown that for any finite point set $\mathbb{R}^d$, any root, and any $\epsilon>0$, there is a Euclidean Steiner $(1+\epsilon,O(\sqrt{1/\epsilon}))$-SLT without any dependence on dimension. We also revisit the core example, designed by Solomon, in the plane and its generalization to $d$-space.

We develop a spectral approach to Sidorenko-type inequalities and apply it to establish sharp edge-spectral supersaturation results. Let $H$ be a bipartite graph with $v$ vertices and $e$ edges, where $v\le e$, and write $M(G)=2e(G)$. We prove that Sidorenko's conjecture is equivalent to a spectral strengthening: $$ \hom(H,G)\ge M(G)^e |V(G)|^{v-2e} \quad \text{ if and only if }\quad \hom(H,G)\ge \lambda(G)^{2e-v}M(G)^{v-e}.$$ We also introduce an operator-norm certificate which, via the Riesz–Thorin interpolation, gives direct proofs of the spectral Sidorenko inequality in several cases. The converse direction in the equivalence theorem is proved by a tensor-power spectral regularization lemma. Our main result provides a unified framework to prove sharp asymptotic edge-spectral supersaturation results for degenerate bipartite graphs with the Sidorenko property, including complete bipartite graphs and even cycles. Let $S_{t-1,m}$ be the split graph with $m$ edges obtained by joining a clique $K_{t-1}$ to an independent set. For every $m$-edge graph $G$ with $\lambda(G)>\lambda(S_{t-1,m})$, $$\texttt{#} K_{t,t}(G) \ge \Big(\frac{2^{-(t-1)^2}}{(t!)^2}-o(1)\Big)m^t \quad \text{and}\quad \texttt{#}C_{2t}(G) \ge \Big(\frac{(t-1)!}{2t^t}-o(1)\Big)m^t.$$ Both constants are best possible: the first is attained asymptotically by random graphs, while the second is attained by split graphs. The supersaturation proofs combine spectral Sidorenko inequalities with a heavy-edge pruning process, a Perron-vector localized/delocalized dichotomy, and incidence-matrix inequalities.

We use the connective formal group law to define a one-parameter ($\beta$-)deformation of the motivic Segre classes of Schubert cells in the $d$-step flag variety. This $\beta$-deformation specializes to the motivic Segre classes of Schubert cells when $\beta=1$ and to the Segre-Schwartz-MacPherson classes of Schubert cells when $\beta=0$ (though the $\beta$-deformations are not simply the $\beta$-homogenizations of the motivic Segre classes). We define rational function representatives for the $\beta$-deformed classes in the $d=1$ case in terms of a solvable lattice model, and we prove a combinatorial formula for the structure constants in the $\beta$-deformed basis in the $d=1$ case using Knutson-Tao puzzles. Unexpectedly, the structure constants are built out of entries of $R$-matrices for the multi-parameter quantum group of type $\widehat{a}_2$ (which suggests an action of multi-parameter quantum groups on equivariant connective $K$-rings of Nakajima quiver varieties, with their convolution algebra structures). We show that our $\beta$-deformations arise from canonical elements in a quotient of the torus-equivariant algebraic cobordism ring of the cotangent bundle of the flag variety by proving that the canonical elements satisfy a GKM type condition (in future work, we will show that these canonical elements also lie in the torus-equivariant connective $K$-ring, in the sense of Karpenko–Merkurjev, of this cotangent bundle).

In 1994, Frieze and Teng proposed an integer linear programming formulation of the NP-Complete Exact Partition problem, whose LP-relaxation they claimed was non-degenerate. Contrary to their claim, we show how an instance of Exact Partition can produce a degenerate polytope, and study conditions for which this can happen. We then give details of one of the smallest such degenerate Frieze-Teng polytopes, along with a closely related non-degenerate Frieze-Teng polytope that encodes an equivalent problem. We note that for the purposes of the complexity results in the literature that use their formulation, these degenerate polytopes can be avoided via a simple preprocessing step.

In 1980, Paul Erdős posed the following problem: For every positive integer $n,$ determine a nonhamiltonian graph of order $n$ having the maximum number of Hamilton paths. We solve the more general problem of determining the nonhamiltonian graphs of order $n$ having the maximum number of paths of length $k$ for given integers $n$ and $k$ with $1\le k\le n-1.$ The case $k=n-1$ gives a solution to Erdős's problem and the case $k=1$ corresponds to a theorem due to Ore and Bondy.

Let $f: T\to \{ 0,1 \}$ be a Boolean function on the Boolean half-slice, $T$, \ie elements of $\{0,1\}^n$ with Hamming weight $n/2$. We show that if $f(x)+f(y)=f(x+y)$ holds with probability $\frac{1+\delta}{2}$ over a uniform pair $(x,y)$ such that $x,y,x+y\in T$, then $f$ agrees with some linear function on at least $\frac{1+\delta}{2}-o(1)$ fraction of the points in $T$. More generally, we show that if $f$ passes the natural $k$-query BLR test with probability $\frac{1+\delta}{2}$ for any $k\geq3$, then it must agree with some affine function at $\frac{1+\delta^{\frac{1}{k-2}}}{2}-o(1)$ fraction of the points in $T$. The only other known linearity test for the slice in the low soundness regime (i.e., when $\delta$ can be arbitrarily small) was given by Kalai, Lifshitz, Minzer, and Ziegler [FOCS'24]. Our result improves upon this result in two significant ways: firstly, it works for $k=3$ queries, instead of requiring $k\geq4$; secondly, our result is sharper, e.g., when $k=4$, we are able to conclude an agreement of $\frac{1+\sqrt{\delta}}{2}-o(1)$ instead of $\frac{1+c\sqrt{\delta}}{2}$ for $c\approx.0035$. In particular, our result matches (up to the $o(1)$ term) the conclusion one obtains over the full hypercube via the classical BLR analysis. Our main technical contribution is a new dense model theorem using bounds on Krawtchouk polynomials. Using these Krawtchouk polynomial bounds, we also obtain a simple $k$-query test ($k\geq 5$) that avoids any use of the dense model machinery. This simplified test naturally extends to the slice over the $q$-ary hypercube, giving the first such result over larger alphabets.

In additive number theory, a finite set $A$ of integers is an $h$-basis for $n$ if every integer in $\{0,1,2,\ldots, n\}$ can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. This paper introduces a new class of problems for these and related additive bases. The problems are designed, in part, to be susceptible to solution by AI.

Let $\mathbb{F}_q$ denote the finite field of order $q$. For $q$ odd, we investigate the planarity over $\mathbb{F}_{q^3}$ of the family $$ f_{E,A,B,C,D}(X) := EX^2+ AX^{q+1}+ BX^{q^2+1}+CX^{2q} +DX^{2q^2}\in \mathbb{F}_{q}[X]. $$ Using results from the theory of q-polynomials, we establish conditions under which these polynomials are planar functions. In particular, we provide characterizations for the planarity property and present new families of planar trinomials, quadrinomials, and pentanomials.

H. Fujimoto showed that for a complete minimal surface in $\mathbb{R}^m$, if the Gauss map is non-degenerate, then it omits at most $\frac{m(m + 1)}{2}$ hyperplanes in the complex projective space $\mathbb{P}^{m - 1}$ in general position, and that the number $\frac{m(m + 1)}{2}$ is best possible for all odd integers $m \geq 3$ and for even integers with $4 \leq m \leq 16$. In this paper, we prove that the number $\frac{m(m + 1)}{2}$ is also best possible for all even integers $m \geq 4$, as conjectured by Fujimoto. The main tool is a special planar network $(\Gamma_0, \omega)$ in the theory of positive matrices.

We introduce modular inequalities for complements of plane curves, based on a Combinatorial Aomoto complex construction associated with the weak combinatorial type of a curve. We use this as a tool to investigate twisted Alexander polynomials, in particular to study the characteristic polynomial of the Milnor fiber associated to a projective plane curve, i.e. the classical Alexander polynomial. We give criteria for the non-triviality of the resonance in degree $1$, over a positive characteristic field, for curves with quasi fiber–type structure and we compute lower bounds for the multiplicities of some roots of twisted Alexander polynomials for this type of curves. We apply these results to theoretically compute the multiplicities of certain roots of twisted Alexander polynomials for fiber–type curves.

Mixed radial Moore graphs are approximations of mixed Moore graphs that preserve the distance-preserving spanning tree for some vertices. One way to measure their resemblance to a mixed Moore graph is using the status measure. The status of a graph is defined as the sum of the distances between all pairs of ordered vertices. Mixed radial Moore graphs with minimum status are closer to mixed Moore graphs according to this measure. The existence of mixed radial Moore graphs is still unknown for most values of the degree and the diameter. In this work, we develop an integer programming model (IP) to find mixed radial Moore graphs of diameter 3 with minimum status. As a result, we show the existence of these graphs for several new values of the degree and the diameter.

We explain a computer enumeration of all partial groups (in the sense of Chermak) of order at most 10. An accompanying dataset contains a full list, consisting of 123,650 partial groups of order at most 9 and 178,937,003 partial groups of order 10; the paper itself contains a complete list of indecomposable partial groups of order at most 5. Inspection of the data led us to conjecture and then prove two results: that indecomposable partial groups of dimension two less than their order are precisely skeleta of groups of that order, and partial groups of (higher Segal) degree at most 2 are 2-coskeletal.

We prove the Erdős Matching Conjecture for $4$-uniform hypergraphs for every matching number $s\ge6004$. Specifically, for integers $s\ge 6004$ and $n\ge 4s+4$, if a $n$-vertex 4-uniform hypergraph $\mathcal F$ does not contain a matching of size $s+1$, then \[ |\mathcal F|\le \max\left\{\binom{4s+3}{4},\binom n4-\binom{n-s}{4}\right\}. \] Our proof introduces a finite-board reduction for general uniformity. It reduces the global conjecture to a lower-uniformity bound and a fixed finite optimization at the two adjacent vertex numbers where the candidate constructions exchange dominance. In the $4$-uniform case the board has $19$ vertices, and its weighted inequality splits into $19$-, $15$-, and $11$-vertex layers. The hardest layer is resolved by exact rational dual certificates and deterministic integer searches. All computer-assisted steps are checked in exact arithmetic by verifiers that reconstruct the finite systems directly from their mathematical definitions.

A set of permutations of $\{1,2,\dots,n\}$ is $t$-intersecting if any two permutations agree on at least $t$ inputs. A recent work by Kupavskii, in the spirit of the Erdős-Ko-Rado Theorem, shows that for all $t\leq n-O\left(\frac{n\log\log n}{\log n}\right)$, every $t$-intersecting family of permutations of $\{1,2,\dots,n\}$ with the maximum size must be isomorphic to the set $$A_k = \{\sigma : \sigma(i)=i\text{ for at least } t+k \text{ indices } i\in\{1,2,\dots,t+2k\}\}$$ for some $k$. By refining Kupavskii's spread approximation technique, we prove that this conclusion holds for a wider range of $t\leq n-n^{5/7+\varepsilon}$.

Motivated by a recent random pipe dream model, we study a family of probability distributions on \(S_n\) arising from Bott–Samelson varieties over finite fields. More precisely, for a word \(R\), we consider the Bott–Samelson map \(\pi_R:\mathrm{BS}^R\to \mathcal{F}\ell_n\) and define a distribution \(\mathbb{P}_{R,q}\) by counting the \(\mathbb{F}_q\)-points in the inverse images of Schubert cells. For a suitable choice of parameters \(p_1=q/(1+q)\) and \(p_2=1/q\), this construction recovers a special case of the random pipe dream distribution. The main problem considered in this note is to determine which combinatorial properties of a reduced word are detected by the distribution \(\mathbb{P}_{R,q}\). We prove the stronger statement that, for arbitrary reduced words \(R_1,R_2\), the equality \(\mathbb{P}_{R_1,q}=\mathbb{P}_{R_2,q}\) as functions of \(q\) holds if and only if \(R_1\) and \(R_2\) lie in the same commutation class. In particular, equality of distributions already forces the two words to represent the same permutation. The proof combines the Bott–Samelson interpretation with Demazure products, commutation-class invariants, and Hecke-algebraic arguments.

We prove that the maximal dimension $d_N$ of an irreducible representation of the symmetric group $S_N$ satisfies $$d_N=\sqrt{N!} \, e^{-(\mathfrak{d}+o(1))\sqrt{N} }, \quad N\to \infty,$$ for some constant $\mathfrak{d}>0$. This answers a question raised by Vershik–Kerov in 1985.

We prove that, for every integer $r\ge 3$, the set $\Pi^{(r)}_\infty$ of Turán densities of (possibly infinite) families of $r$-graphs contains non-degenerate intervals, including an interval of the form $[1-\delta_r,1]$ for some $\delta_{r}>0$. This answers a question of Frankl, Peng, Rödl and Talbot from 2007. This also shows that the Hausdorff dimension of $\Pi^{(r)}_\infty$ has the maximum possible value 1, thus resolving a question of Grosu from 2016, whereas previously it was not even known whether it is non-zero. We also derive that the set of uniform Turán densities of finite families of $3$-graphs is dense in a non-degenerate interval.

Let $\mathbf{G}=\{G_1,\dots,G_{n-1}\}$ be a collection of not necessarily distinct $n$-vertex graphs with the same vertex set $V$. A path $P$ with $V(P)\subseteq V$ and $|E(P)|\leq n-1$ is called rainbow in $\mathbf{G}$, if there exists an injection $\phi\colon E(P)\to [n-1]$ such that $e\in E(G_{\phi(e)})$ for each $e\in E(P)$. The graph collection $\mathbf{G}$ is said to be rainbow panconnected if for every pair of vertices $x,y\in V$, there exists a rainbow path of $k$ vertices joining $x$ and $y$ in $\mathbf{G}$ for every integer $k\in \left[d_{\mathbf{G}}(x,y)+1, n\right]$, where $d_{\mathbf{G}}(x,y)$ is the length of a shortest rainbow path between $x$ and $y$ in $\mathbf{G}$. In this paper, we study the rainbow panconnectivity of $\mathbf{G}$ under the minimum degree condition. Our result improves upon the corresponding results of [J. Graph Theory, 104(2)(2023), 341–359] and [Electron. J. Combin., 32(4)(2025), \#P4.17].

For every fixed integer $t\geq 3$, we construct an $n$-vertex $K_{2,t+1}$-free graph containing $\Omega_t(n^2)$ copies of $K_{t,t}$. Combined with a simple counting argument, this shows that \[ \mathrm{ex}(n,K_{t,t},K_{2,t+1})=\Theta_t(n^2). \] This answers a question of Spiro.

Recently, Ma, Shen and Xie broke the Erdős barrier for off-diagonal Ramsey numbers $R(\ell,C\ell)$, achieving the first exponential improvement over the classical lower bound for every $C>1$ and sufficiently large $\ell$. Hunter, Milojević, and Sudakov later gave a simplified proof using Gaussian random graphs and obtained better quantitative bounds. In this paper we prove a further improvement, and show that the exponent in the Ramsey lower bound can be increased by a strictly positive amount for every fixed $C>1$; as $C\to\infty$, the gain is asymptotically $\Theta(p_C^{-1/2}/\log C)$. The improvement is achieved by replacing the subgaussian estimate for truncated Gaussians with a sharp cumulant generating function bound.

In this paper, we classify connected amply regular graphs with diameter $d \geq 4$ and parameters $(v, k, \lambda, \mu)$ satisfying $\mu = \frac{k-1}{2}$, where $k\geq 5$ is odd. We prove that such a graph must be exactly one of the following: the $5$-cube, the graph $\K_2 \square \Lambda$, where $\Lambda$ is the unique bipartite $(0,2)$-graph on $14$ vertices, or the point–block incidence graph of a group divisible design with the dual property, namely a $GDDDP\left(2, k+1;\, k;\, 0, \frac{k-1}{2}\right)$. For the last family, we give equivalent characterizations in terms of bipartite $Q$-regular graphs and relation graphs of symmetric association schemes with five classes. Furthermore, we present constructions of such amply regular graphs, yielding infinite families of examples derived from Paley graphs, Peisert graphs, and Paley digraphs.

A graph $G=(V,E)$ is said to be word-representable if there exists a word $w$ over the alphabet $V$ such that two distinct letters $x,y\in V$ alternate in $w$ if and only if $xy \in E$. Word-representable graphs form a well-studied graph class with connections to graph orientations, combinatorics on words, and graph coloring. A near-triangulation is a planar graph in which every face except the outer face is a triangle. Several subclasses of near-triangulations have previously been investigated in the context of word-representability, including polyomino triangulations, triangulations of rectangular polyominoes with a single domino tile, $K_4$-free near-triangulations, face subdivisions of triangular grid graphs, triangulations of grid-covered cylinder graphs, and chordal near-triangulations. In this paper, we obtain a complete characterization of word-representable near-triangulations in terms of forbidden induced subgraphs. Our result unifies and extends the previously known characterizations for the above subclasses, while also correcting inaccuracies appearing in earlier works.

The Bruhat order on permutation matrices extends to alternating sign matrices via corner-sum matrices, where the order is given by entrywise domination. A classical result of Lascoux and Schützenberger states that alternating sign matrices form the Dedekind-MacNeille completion of the Bruhat order on permutations. Brualdi and Dahl introduced alternating sign hypermatrices as a three-dimensional analogue of alternating sign matrices and used them to generalise Latin squares, which may be viewed as three-dimensional analogues of permutation matrices. In this paper, in analogy with the two-dimensional case, we define and study a Bruhat order $\preceq_B$ on Latin squares and alternating sign hypermatrices. We introduce the corresponding corner-sum hypermatrices $\mathcal C_n$ and prove that entrywise domination on $\mathcal C_n$ encodes this order. We show that $\mathcal C_n$ is a distributive lattice, but that, unlike in dimension two, it is not the Dedekind-MacNeille completion of the poset of Latin squares. We further characterise the covering relations for $\mathcal C_n$ and prove rank formulae generalising the classical case of alternating sign matrices. Finally, we define monotone hypertriangles, prove that they are in bijection with $\mathcal C_n$, and show that they also encode the order by entrywise domination.

In this work, we investigate the algorithmic aspects of two natural extensions of hereditary classes: the edge-apex class and the edge-add class, recently introduced by Singh and Sivaraman. These are defined as the graph classes obtained by at most one edge deletion or one non-edge addition, respectively, from a hereditary class $\mathcal{G}$. Building on earlier results showing that both classes remain hereditary and admit finite forbidden induced subgraph characterizations whenever $\mathcal{G}$ does, we focus on the Weighted Maximum Clique Problem (WMCP) and the Weighted Maximum Independent Set Problem (WMISP). We first present algorithms for WMCP and WMISP on both the edge-apex and edge-add classes of hereditary graph classes. Extending this framework, we introduce the notion of the $\mathcal{G}$-edge distance of a graph $G$, denoted by $\xi_{\mathcal{G}}(G)$, which quantifies how far $G$ is from the class $\mathcal{G}$ in terms of the minimum number of edge deletions or non-edge additions needed to transform it into a member of $\mathcal{G}$. By parameterizing with respect to this distance, we show that both WMCP and WMISP can be solved in $O^*(2^k)$ time on graphs whose $\mathcal{G}$-edge distance is $k$, provided these problems admit polynomial-time algorithms within the class $\mathcal{G}$. This result extends earlier algorithmic characterizations of the single edge-apex and edge-add classes to the more general setting of $k$-edge-distant graphs. By combining our general results with known properties of transitive graphs, we show that WMCP and WMISP can be solved in $O^*(2^k)$ time for graphs with transitive-edge distance $k$.