Papers

In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Since then, a number of results on truncated theta series have been proved, including truncated Jacobi triple product identity. In this paper, we provide partition-theoretic interpretations for the bilateral truncated Jacobi triple product identity in terms of the minimal excludant integer.

The Ramsey number $r_k(s,m)$ is the smallest $N$ such that any red/blue coloring of the $k$-subsets of $[N]$ contains a red $s$-set or a blue $m$-set. For fixed $k$ and $s$, and for sufficiently large $m$, the tower growth rate is determined by the stepping-up lemma, but for $s=m=k+1$ the available stepping-up lemmas do not apply. Fox asked for estimates of $r_k(k+1,k+1)$. Pudlák, Rödl, and Wesley gave the first tower-type bound: $r_k(k+1,k+1)\ge s_3(\lfloor k/4\rfloor)\ge 4\operatorname{twr}_{\lfloor k/4\rfloor-4}(2)$, where $s_3(k)$ is the $3$-color shift number and $\operatorname{twr}_1(2)=2$, $\operatorname{twr}_{i+1}(2)=2^{\operatorname{twr}_i(2)}$. In this paper, for $k\ge 6$, we improve the lower bound to $r_k(k+1,k+1)> s_3\bigl(\lfloor k/2\rfloor-2\bigr)$ by overcoming an obstruction in their construction. In addition, we give an exact characterization of $s_3(k)$ and, for $k\ge 5$, obtain a new explicit lower bound $s_3(k)\ge(\operatorname{twr}_{k-2}(2))^2$, which improves the result of Pudlák and Rödl. Consequently, for $k\ge 14$, $r_k(k+1,k+1)>(\operatorname{twr}_{\lfloor k/2\rfloor-4}(2))^2$.

For a finite set $A$ of positive integers, let $H(A)$ be its set of subset sums, including the empty sum, and let $g_k(n)$ be the least $N$ for which some $n$-element set $A\subseteq[N]$ has $H(A)$ free of nonconstant $k$-term arithmetic progressions. The problem of determining $g_k(n)$ was posed by Erdős and Sárkőzy. In the three-term case, we prove a lower bound equal to the exact bandwidth of the ternary grid. If $T_m=[x^m](1+x+x^2)^m$ is the central trinomial coefficient, then \[ g_3(n)\ge \frac{T_n-1}{2}+\sum_{j=0}^{n-1}T_j =\left(\frac{\sqrt{3}}{2\sqrt{\pi}}+o(1)\right)\frac{3^n}{\sqrt{n}}. \] For general $k \ge 4$ we show \[ g_k(n)\gg_k \left(\frac{k-1}{k-2}\right)^n n^{-\log_2((k-1)/(k-2))} \] In the opposite direction, a carry-free digit construction based on nearly-regular graphs gives \[ \limsup_{n\to\infty}g_k(n)^{1/n} \le \min_{p\ \mathrm{prime},\ p\ge3}p^{2/(\min\{p,k\}-1)}. \] Consequently, as $k\to\infty$, the logarithm of the lower exponential rate is at least $(1+o(1))/k$, while the logarithm of the upper exponential rate is at most $(2+o(1))\log k/k$.

We show that when the number of zeros or poles is at least four, every connected component of the strata of differentials in genus one with prescribed zero and pole orders is not an orbifold $K(\pi,1)$. For quadratic differentials, this provides infinitely many counterexamples to a conjecture attributed to Kontsevich, as well as to a folklore conjecture concerning the contractibility of spaces of Bridgeland stability conditions.

Let $\mathcal{D}_n$ denote the average number of passes of the stack-sorting map $s$ required to sort a permutation in $S_n$. We use the recently introduced framework of stack-sorting diagrams and tableaux to prove that the limit $\lim_{n\to\infty}\mathcal{D}_n/n$ exists. This resolves a longstanding conjecture of West originally proposed in $1990$. As a consequence, we also provide a monotonically increasing sequence that converges to $\lim_{n\to\infty}\mathcal{D}_n/n$, improving upon Defant's lower bound of $\lambda\approx 0.62433$.

We compare spanning-tree counts at a fixed number of vertices for Cartesian products of paths $P_r$, $r\ge1$, or cycles $C_s$, $s\ge3$. With the number $d$ of factors fixed, an equal-volume tuple that is more balanced in the logarithmic majorization order has at least as many spanning trees, with equality only for permutations. This comparison persists after adjoining any fixed finite connected Cartesian factor; equal-sided blocks therefore uniquely maximize the count at admissible block volume $n^d$. We prove the comparison through strict log-convexity of one-dimensional heat traces in logarithmic length. The Hartman–Watson representation expresses the cycle interpolation as a positive mixture of theta functions, and the path case follows from it. The same heat-trace order decreases every positive inverse-power sum of the nonzero Laplacian eigenvalues, and hence the total effective resistance. Cylinders behave differently: square balancing can fail. For $P_r\square C_s$ of area $N=rs$, extending the product formula to real height and optimizing over integer circumferences $3\le s\le N$ yields an optimum on the scale $N^{1/3}$. Along areas having a divisor asymptotic to that optimum, quantitative estimates place every graph-maximizing circumference on the same scale. For two-dimensional tori, balancing remains optimal, but the gain in the logarithmic spanning-tree count admits no positive uniform lower bound proportional to the decrease in the side sum.

The forest polynomials $\mathfrak{P}_a$ of Nadeau-Tewari form a $\mathbb{Z}$-basis of $\mathbb{Z}[x_1, x_2, \dots]$ whose role for the cohomology of the quasisymmetric flag variety parallels that of Schubert polynomials for the classical flag variety. Nonnegativity of the structure constants $\beta^c_{a,b}$ in $\mathfrak{P}_a \mathfrak{P}_b = \sum_c \beta^c_{a,b} \mathfrak{P}_c$ is known, but no Littlewood-Richardson-style enumerative rule has been available. We give such a rule: $\beta^c_{a,b}$ counts pairs of forest RC graphs of forest-codes $a$ and $b$ whose lift product lands on a forest RC graph of forest-code and weight both equal to $c$. The same rule descends to the cup product on $H^\bullet(QFl_n)$. The proof introduces a Schubert bialgebra $\mathcal{A}$ and lifts the multiplication on its graded dual $\mathcal{D}$ to a product on a free abelian group $\mathcal{B}RC$ of bounded RC graphs; the same machinery yields enumerative LR rules for the dual Schubert, dual key, dual forest, and dual slide bases of $\mathcal{D}$.

We prove that a biased graph is gainable over the group $\mathsf{Z}_{3}$ if and only if it contains no minor isomorphic to $(4K_{2},\emptyset)$, $\pm K_{3}$, or $-K_{4}$. We develop a theory of "partial groups" that is analogous to that of partial fields, and we use this theory to show that a biased graph is gainable over every non-trivial group if and only if it is gainable over $\mathsf{Z}_{2}$ and $\mathsf{Z}_{3}$. From this we derive an independent proof of the theorem due to Gerards that a biased graph is gainable over every non-trivial group if and only if it has no minor isomorphic to $(3K_{2},\emptyset)$, $\pm K_{3}$, or $-K_{4}$.

An edge-coloring of a graph $G$ is called a strong edge-coloring if all its color classes are induced matchings in $G$; the minimum number of colors required for such a coloring, denoted by $\chi_{s}'(G)$, is known as the strong chromatic index of $G$. For each vertex $v$ of a graph $G$, let $d_G(v)$ denote the degree of $v$ in $G$. Let $G$ be a bipartite graph with partite sets $A$ and $B$, and let $\Delta_A=\max\{d_G(a): a\in A\}$ and $\Delta_B=\max\{d_G(b): b\in B\}$. A conjecture of Brualdi and Quinn Massey asserts that \( \chi_s'(G) \le \Delta_A \Delta_B\). In this paper, we show that \(\chi_s'(G) \le 1.676\, \Delta_A \Delta_B\) provided that the product $\Delta_A\Delta_B$ is sufficiently large.

We study the number $\Nu(\Per(G_{\lambda}))$ of distinct monomials with nonzero coefficients in the group permanent of an abelian $p$-group $G_\lambda$ associated with a partition $\lambda$ of a positive integer $N$. First, we derive an explicit formula for $\Nu(\Per(G_{\lambda}))$ in terms of the partial column sums of the Young diagram of $\lambda$. Next, we show that the relative order of the values $\Nu(\Per(G_{\lambda}))$ is determined by a lexicographic comparison of the conjugate Young diagrams. Finally, we investigate congruence properties of $\Nu(\Per(G_{\lambda}))$ for abelian $p$-groups and establish a criterion involving Wolstenholme primes.

It is shown that there exists $c > 0$ so that the Cayley graph over any finite abelian group $Z$ generated by $c \log |Z|$ random elements is properly 3-colorable with high probability (as $|Z| \to \infty$). This is asymptotically tight and improves the best-known bound due to Alon of $\frac{1}{4}\log \log |Z|$ elements. It also settles the abelian case of Alon's suggestion that a bound of $c \log |G|$ may hold for any finite solvable group $G$.

We study the trinomial equation $x^n +px +q =0$. Here $p$ and $q$ are both real and nonzero. For $n\ge3$, expressions for the roots have been published as hypergeometric series in powers of the parameter $q^{n-1}/p^n$. For the special case of the cubic ($n=3$), we employ Kummer's identities to derive alternative series solutions in powers of the discriminant $D$, and also series in powers of $1/D$. We next derive new series, in powers of $D$ and also in powers of $1/D$, for all $n\ge 3$. The resulting series suggest identities analogous to Kummer's identities, for higher order hypergeometric series.

We resolve Erdős Problem 550, originally asked as question (2) of Erdős, Faudree, Rousseau, and Schelp. Precisely, for fixed integers $k\geq 2$ and $1\leq m_1\leq \cdots \leq m_k$, we prove that, for every sufficiently large $n$ and every $n$-vertex tree $T$, $R(T,K_{m_1,\ldots,m_k}) \leq (k-1)(R(T,K_{m_1,m_2})-1)+m_1$. The proof combines an off-Turán tree-embedding theorem, proved by regularity and whole-edge allocation, with a compactness theorem for bounded-rank hypergraph obstructions. The full and unconditional proof has been formally verified in Lean.

A signed graph $\Sigma=(G,\sigma)$ is a graph $G$ with edges given signs $1$ or $-1$ defined by the function $\sigma$. The adjacency matrix of $\Sigma$ is defined as per these signs. The relation between the largest eigenvalue of $G$ and $G-v$ has been studied in recent years, where $G-v$ is the graph obtained from $G$ by deleting the vertex $v$. In 2020, Sun and Das proved that the difference of the squares of the largest eigenvalues of the graphs $G$ and $G-v$ is bounded above by $2d(v)-1$ where $d(v)$ is the degree of $v$. A similar result need not be true for the largest eigenvalue of signed graphs. In this paper, we prove that the result is valid for the spectral radius of signed graphs. On the other hand, the signed graph version of Hoffman's chromatic number bound was proved by Wang et al. in 2021. They also discussed the difficulty in proving the extended version encompassing all eigenvalues of $\Sigma$ as was done for unsigned graphs by Wocjan and Elphick. We note down a consequence of Wocjan and Elphick's lower bound for the chromatic number in terms of all the eigenvalues of $\Sigma_+$; all the eigenvalues of $\Sigma$ and $\Sigma_-$, where $\Sigma_+$ (resp. $\Sigma_-$) is the spanning subgraph induced by the positive (resp. negative) edges. We give examples where the result fails even under various restrictions on the signed graph. Finally, we improve an upper bound for the $k$-th power of the largest eigenvalue given by Stanić in terms of walks in signed graphs and give lower bounds for the least eigenvalue in terms of various parameters of $\Sigma_+$ and $\Sigma_-$.

A left-regular bipartite graph $G$ of degree $d$ is called a $(t,\alpha)$-small-set-expander if every subset $X$ of left vertices of size at most $t$ has at least $\alpha |X|$ neighbors. Such a graph is an optimal small-set expander if small subsets have as many neighbors as possible. We characterize optimal expanders combinatorially via girth and prove the existence of $s$-optimal expanders for every $s$. We also prove that $s$-optimality yields new "transfer" lower bounds on the number of neighbors of sets of size $h\geq s$. Finally, as an application, we discuss the use of optimal small-set expanders in building good codes for key exchange protocols in post-quantum cryptography.

Continuing our previous work, this paper closely studies the relationship between the cluster algebra structures on the coordinate ring of Schubert cells and those on the coordinate ring of partial flag varieties. We give a finite-type classification for these cluster structures and point out several results that were left open in our previous work.

In the 1980s, Erdős and, independently, Frankl and Pach conjectured that, for sufficiently large $n$, every $(d+1)$-uniform family on $\{1,\ldots,n\}$ with VC-dimension $d$ has size at most $\binom{n-1}{d}$, the size of a star. Ahlswede and Khachatrian disproved this conjecture in 1997 by giving a family of size $\binom{n-1}{d}+\binom{n-4}{d-2}$. This value has since been widely believed to be best possible, and Mubayi and Zhao explicitly conjectured its optimality in 2007. Very recently, Wang, Xu and Zhang proved their conjecture for $d=2$ and $n\ge 7$, providing further support for this belief. Surprisingly, we show that the Mubayi-Zhao conjecture is false for every $d\ge 3$ by constructing families larger than the Ahlswede–Khachatrian bound. Our constructions suggest that the answer to the Erdős–Frankl–Pach problem depends delicately on both $n$ and $d$.

Ferdinand Ihringer proved that Boolean degree one functions on the Grassmann scheme $J_q(n,k)$ are trivial when $\min(k,n-k) \ge 2$ and $n$ is large enough. We provide a mostly self-contained exposition of this result. All results were formalized by ChatGPT 5.6 Sol, and are available on GitHub at YuvalFilmus/grassmann-degree-one.

We present an algorithm for exact-size sampling of uniformly random Pólya trees with linear expected runtime.

A convex splinter $K$ is a union of convex sets in $\mathbb{R}^d$ such that every minimal affine dependent set of $\mathbb{R}^d$ contained in $K$ is contained in one of the sets. The study of convex splinters was motivated by the study of flat transversals to convex sets. We extend several variations of Helly's theorem from convex geometry to convex splinters. These include fractional and colorful variations of Helly's theorem. We also extend Tverberg's theorem to convex splinters.

Rigo and Salimov (2015) proved that the number of binary \(2\)-binomial equivalence classes of words of length \(n\) is the \(n^{\text{th}}\) cake number. We give a geometric explanation of this identity by constructing an explicit arrangement of \(n\) planes in three-dimensional space whose chambers are naturally indexed by these equivalence classes. This arrangement is the three-dimensional member of an infinite family of hyperplane arrangements. In dimensions \(1\), \(2\), and \(3\), the corresponding quotients recover abelian equivalence, a natural intermediate equivalence between abelian and \(2\)-binomial equivalence, and binary \(2\)-binomial equivalence itself. In higher dimensions, the same family realises natural refinements of \(2\)-binomial equivalence. We also determine the sizes of the resulting classes. Each size is given by a coefficient of a suitable Gaussian binomial coefficient. This yields the full class-size distribution for binary \(2\)-binomial equivalence and stabilisation results for the number of classes of any fixed cardinality.

We prove Cusick's conjecture on the binary sum-of-digits function. More precisely, for every integer \(t\ge 1\) we show that \[ c_t:=\lim_{N\to\infty}\frac{1}{N} \#\{0\le n<N:\ s_2(n+t)\ge s_2(n)\}>\frac{1}{2}, \] and in fact obtain the explicit bound \[ c_t\ge \frac{1}{2}+2^{-2s_2(t)-1}, \] where \(s_2(m)\) denotes the number of ones in the binary expansion of \(m\). The proof is based on an exact deconvolution which replaces the distribution of \(s_2(n+t)-s_2(n)\) by a finite stopped random-walk law. The required bias is then proved through first-exit medians for principal subsequence ideals.

We study the fixed-density semi-inducibility profiles of the red-blue star $S_{2,1}$, which has one distinguished center, two red edges and one blue edge. For an $n$-vertex graph $G$, let $N(S_{2,1},G)$ be the number of injective labeled copies in which the two red edges of $S_{2,1}$ are mapped to edges of $G$ and its blue edge is mapped to a non-edge of $G$, that is, \begin{align*} N(S_{2,1},G)= \sum_{v\in V(G)} d(v)(d(v)-1)(n-1-d(v)). \end{align*} For every fixed red edge density $\beta\in[0,1]$, we determine both extremal $S_{2,1}$-densities. On the upper side, we prove the missing low-density range and, together with the theorem of Balogh, Lidický, Mubayi, Pfender and Volec for $\beta\ge 1/4$, obtain the full four-branch profile predicted in their work. On the lower side, we show that the natural endpoint profile coming from the quasi-star and quasi-clique constructions is not universal; the correct minimum is given by a one-parameter three-class complement-split family. The proofs use a transfer argument with degree-square tie-breaking, reducing the extremal analysis to almost-regular, threshold and finite-staircase optimizations.

Two families $\mathcal{A}, \mathcal{B} \subset \binom{[n]}{k}$ are cross-intersecting if $A \cap B \ne \emptyset$ for all $A \in \mathcal{A}$ and $B \in \mathcal{B}$, and non-trivial if neither $\m A$ nor $\m B$ is a star. Pyber proved that any two cross-intersecting families $\mathcal{A}, \mathcal{B} \subset \binom{[n]}{k}$ satisfy $|\mathcal{A}||\mathcal{B}| \le \binom{n-1}{k-1}^2$, and the maximum is attained by two full stars. Frankl, as well as Frankl and Wang, conjectured that the sharp bound, when both families are required to be non-trivial, is $h(n,k)^2$, where $h(n,k) = \binom{n-1}{k-1} - \binom{n-k-1}{k-1} + 1$, the size of the Hilton–Milner family. The cases $k=3$, and the range $k\ge8$ and $n\ge 4k$, were established earlier by Frankl and by Frankl and Wang, respectively. In this paper, we prove their conjecture in the full range. We show that every non-trivial cross-intersecting pair $\mathcal{A}, \mathcal{B} \subset \binom{[n]}{k}$ with $n \ge 2k$ and $k \ge 3$ satisfies $|\mathcal{A}||\mathcal{B}| \le h(n,k)^2$. Moreover, we characterize all extremal pairs. Whereas the corresponding sum problem admits asymmetric and unbalanced extremizers, the product extremum forces a balanced, symmetric-or-dual structure: the two families are isomorphic when $n>2k$ and complement-dual when $n=2k$. Independently and contemporaneously with the present work, Frankl and Wang obtained the same bound for $k\ge8$ and $n\ge2k+1$ by a different method. Our proof combines a diversity technique with several new properties of an extended shift operation. Moreover, we show that the problem behaves differently for different uniformities, exhibiting new extremal configurations. In particular, we disprove a related conjecture proposed by Frankl and Wang.

Cheong, Goaoc and Holmsen conjectured that every connected component of the space of line transversals to a family of pairwise disjoint open convex sets in $\mathbb{R}^d$ is acyclic. We disprove this conjecture by showing that the homology may be nontrivial in any fixed dimension provided that $d$ is large enough. More precisely, we show that for every $n \geq 1$ there is a finite family of pairwise disjoint open convex sets in $\mathbb{R}^{3n}$ such that the $(n-1)$st homology (over an arbitrary ring) of the space of line transversals to this family is nonzero.

Menger's theorem on $A$–$B$-paths and Gallai's theorem on $A$-paths are among the most useful results in structural graph theory. Many variants and extensions are known. We add to this line of research and prove results that relate the maximal number of vertex-disjoint paths between vertex sets and a wall to the minimum number of vertices meeting all these paths. We also include types of paths that start and end in the wall.

We investigate standard polynomials for principal subalgebras of path algebras. First, we use standard polynomials to study the $PI$-theory of principal subalgebras. Then we describe the $St_2$-elements and $St_3$-elements of principal subalgebras, giving a characterization of their centers and 3-centers. In addition, we apply these results to combinatorics on words of formal languages, obtaining some explanations from a combinatorial perspective.

Sidorenko's conjecture asserts that every bipartite graph has at least the expected homomorphism density in every graph of a given edge density. Motivated by Cayley-type formulations of Sidorenko-type inequalities, we study a two-sided correlation construction on finite groups. Let $\Gamma$ be a finite group and let $f:\Gamma\to\mathbb{R}$ be a real-valued function. We define a directed kernel on $\Gamma$ by $$\mathcal C_f(x,y)=|\Gamma|^{-1}\sum_{a_1,a_2\in\Gamma:\, xa_1=a_2y} f(a_1)f(a_2)=\mathbb{E}_{z\in\Gamma} f(x^{-1}z)f(zy^{-1}).$$ When $f=\mathbf{1}_A$, this is the normalized size of the intersection $xA\cap Ay$. We prove that, for every finite directed graph $F$, $$t(F,\mathcal C_f)\geq t(\overrightarrow{K_2},\mathcal C_f)^{e(F)}=(\mathbb{E}_{g\in\Gamma}f(g))^{2e(F)}.$$ Equivalently, if $W_f^\times(x,y)=f(xy)$ is the directed product Cayley kernel on $\Gamma$, then the directed $1$-subdivision of every finite directed graph satisfies the same homomorphism-density lower bound in $W_f^\times$.

We introduce the externally supported independence number $\alpha_{\rm es}(G)$ of a graph $G$ as the maximum cardinality of an independent set $B$ with an additional condition, that vertices from $N(B)$ are dominated by vertices in $V(G)-N[B]$. This parameter yields an improved upper bound on the isolation number $\iota(G)$. We show that computing $\alpha_{\rm es}(G)$ is NP-hard, while for trees we present a linear-time algorithm. We also establish several sharp bounds on $\alpha_{\rm es}(G)$ for general graphs, with additional refined results for trees. In several cases, we completely describe the extreme graph classes attaining these bounds.

Determining the conditions under which the direct product of graphs $G$ and $H$ satisfies $\mathrm{Aut}(G\times H)=\mathrm{Aut}(G)\times\mathrm{Aut}(H)$ has been a problem of considerable interest since Sabidussi's classic work in the 1950s. We call such a pair $(G,H)$ stable, and unstable otherwise. Although much progress has been made for graph pairs, the general digraph case has remained completely open. In this paper, we initiate the study of the stability of digraph pairs, and then focus on the stability of a single digraph $G$. This is defined as the stability of the pair $(G,K_2)$ and has been studied extensively when $G$ is undirected. We establish a necessary and sufficient condition for a connected digraph to be unstable, and use it to derive four sufficient conditions for circulant digraphs to be unstable. Moreover, we prove the nonexistence of nontrivially unstable finite arc-transitive circulant digraphs and nontrivially unstable Cayley digraphs of abelian groups of odd order.