Papers

We find some significant special cases of subtraction games with infinite subtraction sets, whose outcome sequences are periodic, but whose SG value sequences are bounded and aperiodic.

A Delannoy–Steinhaus triangle is obtained from a finite sequence by a recurrence governed by the Delannoy numbers. We introduce this construction over $\mathbb{Z}/2\mathbb{Z}$ and study its weight distribution. The relevant Delannoy coefficients are all odd, which reduces every entry to the parity of a consecutive interval of the generating sequence. Encoding these interval parities by prefix parities yields a weight formula depending only on the numbers of zeros and ones in the prefix-parity sequence. We use this formula to determine the complete weight spectrum and the exact multiplicity of each weight. As a consequence, we characterize and enumerate the balanced triangles: a balanced triangle generated by a binary sequence of length $n$ exists if and only if $n+1$ is a perfect square. We also determine the canonical-vector weights, the minimum nonzero weight, the {second-smallest nonzero weight}, the maximum weight, and the average weight.

In this paper, we investigate Chocolate Games. In Chocolate Games, each player cuts a chocolate bar into two chocolate bars and eats one of them so that they do not eat an indicated bitter block. The player who eats the bitter part loses the game. This game can be considered as a generalization of Nim, and previous studies consider the condition that for what kind of the shape of chocolate bar, the Sprague–Grundy value of the position can be calculated by Nim–sum (XOR) of the width and height of the chocolate bar. Higher dimensional cases were also studied. In this paper, we show that the “Tasty condition,” which was introduced in previous work for increasing staircase chocolate bars, can also be used for various other shapes of chocolate bars as a sufficient condition, or in some cases a necessary and sufficient condition, for the Sprague–Grundy value to be given by the Nim-sum of the sizes of the chocolate bar in each dimension.

The maximum size of a weak $P$-free family $\mathcal{F}\subseteq 2^{[n]}$ is denoted by $La(n,P)$. Let $e(P)$ denote the maximum integer $k$ such that the union of any $k$ consecutive layers of $2^{[n]}$ is weak $P$-free. In recent years, multiple examples of posets with $e(P)<\pi^-(P):=\liminf_{n\to\infty} \frac{La(n,P)}{\binom{n}{\lfloor \frac{n}{2}\rfloor}}$ have been found. We add several further posets with $e(P)=1$ to this list. We define a family $\mathcal{F}\subseteq 2^{[n]}$ of size at least $(1.22+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$ that is weak $O_6$-free, where $O_6$ is the six-element crown poset. We also show an infinite set of posets $P$ with $1=e(P)<\pi^-(P)$ that are minimal with respect to this property. Finally, we consider how far apart $e(P)$ and $\pi^-(P)$ can be. We prove that for every fixed finite poset $P$ with $e(P)=1$, there is a constant $\delta_P>0$ such that $La(n,P)\le(2-\delta_P+o(1))\binom{n}{\lfloor n/2\rfloor}$. The value 2 is optimal: explicit vertex-edge incidence posets with $e(P)=1$ have $\pi^-(P)$ values tending to $2$. In contrast, for every $K>0$ we construct a finite poset $P$ with $e(P)=2$ and lower density greater than $K$.

The Tchoukaillon array is an infinite array of the positive integers, arising from a one-row Mancala solitaire, in which each positive integer occurs exactly once. Its zeroth column is the Flavius Josephus sieve and its zeroth row is the sequence of Tchoukaillon numbers; the asymptotics of these two edges are classical results of Andersson and of Broline and Loeb. On the basis of numerical evidence, N. Beluhov conjectured (as relayed by Knuth) that the general entry $T_{i,j}$ satisfies $T_{i,j} \approx (\pi i+2j)^2/(4\pi)$ as $i,j \to \infty$. We prove this conjecture. In fact we establish the stronger uniform estimate $T_{i,j} = (\pi i+2j+2)^2/(4\pi) + O((i+j+1)^{4/3})$, in which both constants $\pi$ and $2$ are produced by the array's own recursion through a Wallis product, independently of the two edge theorems. Equivalently, the square root of the entry is asymptotically linear, $\sqrt{T_{i,j}} = (\sqrt{\pi}/2)\, i + (1/\sqrt{\pi})(j+1) + O((i+j+1)^{1/3})$, the linear blend of the two edge growth-rates. As corollaries we obtain that the level regions $\{T_{i,j} \le V\}$ are triangles up to a boundary of width $O(V^{1/6})$, and an $O(\sqrt{M})$ algorithm that locates the row and column of a given integer $M$.

For every finite-rank crystallographic affine Coxeter system $(W,S)$ and Coxeter element $c$, we construct a minimal linear graded free resolution of the trivial module over $k[M([1,c]_T)]$ supported on a rectified exceptional cluster complex. Hence the affine dual braid monoid algebra is Koszul over every field $k$. The exceptional complex is introduced to recover the principal-fibre topology missing from the direct Reading–Stella labelling. Half-orbit rectification replaces the transjective root labels by ordinary exceptional modules, so that a face $F$ determines an exceptional wide subcategory and the intrinsic weight \[ \omega(F)=\operatorname{cox}(\operatorname{wide}\langle F\rangle). \] The resulting principal fibres are induced subcomplexes and split canonically as joins of subcomplexes attached to connected Dynkin and affine blocks; these subcomplexes are contractible. Affine non-lattice divisibility creates the genuinely nonprincipal case. The McCammond–Sulway completion shows that whenever no greatest interval right divisor exists, all maximal interval right divisors share a common nontrivial complete finite Coxeter component. In the associated exceptional-wide decompositions, this common Coxeter component is the Coxeter element of a Dynkin block, and the subcomplex attached to that block occurs as a common contractible join factor. Thus every nonidentity fibre is contractible, and the weighted-face complex is exact, minimal and linear. In particular, $\operatorname{Tor}^{A_c}_q(k,k)$ is indexed by $q$-vertex exceptional cluster faces in internal degree $q$, and $\operatorname{pd}_{A_c}k=|S|$.

Motivated by an analogue of pancyclicity, we study minimum-degree conditions ensuring that a $2$-connected graph $G$ of order $n$ contains a $2$-connected subgraph of every order $\ell\in\{4,5,\ldots,n\}$. Yin and Wu [A minimum degree condition for a 2-connected graph containing all possible orders of 2-connected subgraphs, Discrete Appl. Math. 387 (2026), 129-136] initiated the study of this problem and showed that the condition $\delta(G)\ge \lceil n/3\rceil+1$ is sufficient. Kashima conjectured that the condition $\delta(G)\ge \sqrt{3n}$ is sufficient. In this paper, we prove that every $2$-connected graph $G$ of order $n$ with $\delta(G)\ge 2n^{2/3}+6n^{1/3}+2$ contains a $2$-connected subgraph of every order from $4$ to $n$. In particular, this gives the first sufficient minimum-degree condition of sublinear order in $n$.

We study two open path problems with prescribed endpoints posed by Bang-Jensen and Gutin. The first asks for a longest $(x,y)$-path in a semicomplete digraph. The second asks whether a locally semicomplete digraph has a Hamiltonian $(x,y)$-path. For semicomplete digraphs, we solve the first problem when the endpoints lie in different strong components. We also prove that if a non-Hamiltonian longest $(x,y)$-path omits a set of vertices, then these vertices together with $x$ and $y$ have a Hamiltonian $(y,x)$-path. This gives an equivalent cycle problem. We then give an exact algorithm that runs in polynomial time when the number of omitted vertices is fixed. For locally semicomplete digraphs, we determine the possible endpoints in the connected nonstrong case. Known results then leave only strong, nonsemicomplete, non-$4$-strong digraphs unresolved. Every such digraph of order at least five has a strong vertex cut of size at most three. Two examples show that a spanning directed path together with a vertex-disjoint directed cycle is not sufficient, and that vertices of one strong component need not occur consecutively on a Hamiltonian path.

We study algebras with straightening laws on join- or meet-semidistributive lattices. We show that for a join-semidistributive (resp. meet-semidistributive) lattice, meet-distributivity (resp. join-distributivity) and Cohen–Macaulayness are equivalent, and that integrality implies these conditions. Thus, Hibi's conjecture that every integral lattice is Cohen–Macaulay holds for join- or meet-semidistributive lattices. For semidistributive lattices, distributivity, integrality and Cohen–Macaulayness are equivalent.

We establish a Newton-type identity for the queer Lie superalgebra $\mathfrak q_N$, relating Sergeev's odd cyclic central elements to Nazarov's one-row Capelli elements. The identity is obtained by comparing Ivanov's generating function for factorial Schur $Q$-functions with the queer Perelomov-Popov product of Grigoryev and Nazarov. Its coefficient expansion yields a triangular change of generators between the odd cyclic and odd one-row families. In particular, the odd one-row Capelli elements generate the center, while the even one-row elements are redundant. In fixed rank, we derive determinantal relations and a generic reconstruction theorem. The basic cyclic Hankel determinant is identified with a resultant and factored into the failure-of-strong-typicality and shifted-resonance factors. After localization at this determinant, the center is generated by the first $2N$ odd cyclic elements; consequently, generic central characters are determined by their values on these elements.

The Schubert vanishing problem asks whether the single Schubert coefficients $c_{u,v}^w$ are zero. In this paper, we consider the non-vanishing problems of double Schubert coefficients $c_{u,v}^w(t)$ and triple Schubert coefficients $c_{u,v}^w(t;y)$. We show that the non-vanishing of $c_{u,v}^w(t;y)$ is completely determined by the non-vanishing of single Schubert coefficients. As a byproduct, we obtain the saturation property of the triple Littlewood–Richardson coefficients $c_{\lambda,\mu}^\nu(t;y)$. Moreover, we pose a conjecture asserting that the non-vanishing of $c_{u,v}^w(t)$ is also determined by the non-vanishing of single or triple Schubert coefficients. We prove a one-side inclusion of the conjecture. For the reverse inclusion, we show that the conjecture holds for the following three cases: the Pieri case, the separated descents case, and the inverse Grassmannian case.

It was known at least to L.E. Dickson in 1901 that $\mathrm{SL}(2,q)$, in its natural action on $\mathbb{F}_q^2\setminus\{0\}$, has a sharply transitive subgroup only when $q\in\{2,3,5,7,11\}$. For $q$ prime, this result stems from Galois' letter to Chevalier in 1832. We extend this result to sharply transitive subsets of $\mathrm{SL}(2,q)$ and show that they only exist when $q\in\{2,3,5,7,11\}$.

In this paper, we investigate the underlying algebraic and combinatorial structures governing the coset representatives (shift vectors) for Construction A of lattices, with a particular focus on equations (14) and (15) presented in Paragraph 3, Chapter 8 of the seminal work by J.H. Conway and N.J.A. Sloane, "Sphere Packings, Lattices and Groups". These dual equations define the boundary conditions for the analytical generation of the 64-dimensional Quebbemann lattice (Q64). We prove that seeking non-zero solutions constrained by arithmetic or geometric progressions yields a structural collapse to the trivial zero vector due to the transcendental nature of {\pi}. Conversely, by relaxing these bounds to unique coordinate configurations, we uncover an exact, closed-form algebraic core governed by the alternating coefficients of the Pascal triangle. Furthermore, we implement an energy-minimization model via continuous-to-discrete projection that yields an optimal shift vector with a strictly integer Euclidean norm ||z||^2 = 20.000000. Finally, we bridge this formulation to the Repeated Differences paradigm of Craig's lattices Anm, showing how the spectral components of the {\Theta}-series are naturally filtered by these binomial structures.

For a prime $p\geq 3$, a somewhat restricted $3$-AP in $\mathbb{F}_p^n$ is a triplet $(x,x+a,x+2a)$, where $x\in\mathbb{F}_p^n$ and $a\in \{0,1,2\}^n$. We prove a counting lemma for somewhat restricted $3$-APs in dense sets in $\mathbb{F}_p^n$. More precisely, we prove that for all $\alpha>0$, there exists $\beta>0$, such that for sufficiently large $n$, if a set $A\subseteq \mathbb{F}_p^n$ has density at least $\alpha$, then it contains at least $\beta$ fraction of all somewhat restricted $3$-APs. Our proof builds on recently developed machinery from [Bhangale, Khot, Minzer, 2026]. Our main new ingredient is an arithmetic regularity lemma for patterns such as somewhat restricted 3-APs. This result is in the spirit of arithmetic regularity lemmas from the theory of Gowers uniformity norms [Green, Tao, 2010] and may be of independent interest.

We initiate a systematic study of Inc-invariant chains of graphs, the combinatorial counterparts of Inc-invariant chains of edge ideals arising in the theory of equivariant Noetherianity. Such a chain consists of graphs on growing vertex sets whose edge sets are compatible with the action of the monoid of strictly increasing maps on the positive integers. We show that several associated combinatorial invariants exhibit rigid asymptotic behavior. The independence number eventually stabilizes, and every fixed entry of the $f$-vector and the $h$-vector of the independence complex is eventually linear. For clique complexes, every fixed entry of the $f$-vector is eventually polynomial, whereas the entries of the $h$-vector are eventually quasi-polynomial. Moreover, the clique and chromatic numbers are eventually quasi-linear, and their difference is eventually at most one. We also prove that the matching number eventually attains the maximal value $\lfloor n/2\rfloor$. Finally, admissible and minimal paths eventually have lengths at most $3$ and $5$, respectively, and their maximal lengths stabilize. These results reveal strong asymptotic regularity in graph families governed by increasing symmetry.

For a graph $G$, let $a(G)$ be the maximum number of vertices in an induced forest. The Albertson-Berman conjecture, posed in 1979, asserts that every $n$-vertex planar graph satisfies $a(G)\ge n/2$. Borodin's bound $a(G)\ge 2n/5$ remains the general lower bound toward this problem. We disprove the conjecture with an explicit 31-vertex plane triangulation $T$ satisfying $a(T)=15$. Moreover, for every integer $k\ge2$, we construct a simple planar graph $M_k$ with $|V(M_k)|=31k$ and $a(M_k)=15k$, so that $a(M_k)/|V(M_k)|=15/31<1/2$. Every member of the family has minimum degree five. The construction starts from a $31$-vertex seed obtained by substituting a $14$-vertex two-terminal gadget into a pentagonal bipyramid, and then uses annular joins along facial triangles to preserve the exact ratio. The resulting graphs are sphere triangulations, and hence maximal planar.

Given a simple graph $G=(V,E)$ of order $p$ and size $q$, a bijection $f : V\cup E \to \{1, 2, \ldots, p+q\}$ is a local total neighborhood antimagic labeling of $G$ if the induced vertex coloring has the property $f^+_{tn}(u) \ne f^+_{tn}(v)$ for every two adjacent vertices $u$ and $v$ where $f^+_{tn}(u) = \sum (f(ux) + f(x))$ over every neighbor $x$ of $u$. The local total neighborhood antimagic chromatic number of $G$, denoted $\chi_{ltna}(G)$ is the minimum number of distinct induced colors over all local total neighborhood antimagic labeling of $G$. In this paper, we determine the local total neighborhood antimagic chromatic number of the join of graphs with distinct parity orders.

For a graph $G$, let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of its positive and negative adjacency eigenvalues. We determine all equality cases in the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies \[ \min \{s^+(G),s^-(G)\}\ge n-1. \] Namely, equality for $s^+$ holds exactly for trees, whereas equality for $s^-$ holds exactly for trees and complete graphs. The proof combines the $P_3$-removal lemma in the no-cut-vertex case with a detailed equality analysis of the underlying doubly nonnegative matrix inequality. Every block is forced to be complete, and a minimal-counterexample argument gives an exact rank-one decomposition of the folded matrix $M^c$. The resulting non-edge vanishings, together with $AX=XA$, rule out an interface between a bridge and a nontrivial block.

Over a decade ago, Koolen, Hayat, and Iqbal posed the problem of whether the distance spectrum determines bipartiteness within the class of connected graphs. In this paper, we resolve this problem in the negative: we explicitly construct an infinite family of counterexamples, where each pair comprises a connected bipartite graph and a connected non-bipartite graph with equal distance spectra.

We prove that each $2$-edge-coloured spanning subgraph $G$ of $K_{n,n}$ with $\delta(G)\ge \lfloor (2n+1)/3 \rfloor$ can be covered by at most three monochromatic components. We provide a $2$-edge-coloured spanning subgraph of $K_{n,n}$ showing this minimum degree condition is sharp.

Let $Q$ be a nonzero $s\times t$ $(0,1)$-pattern, and let $m\ge s$ and $n\ge t$. An $m\times n$ matrix is strongly $Q$-forcing if every $1$-entry belongs to an $s\times t$ submatrix equal to $Q$. Let $F^{*}(m,n,Q)$ count these matrices. Put $H=m-s+1$ and $W=n-t+1$. We prove \[ F^{*}(m,n,Q)\ge 2^{HW}. \] Writing $r$ and $c$ for the numbers of nonzero rows and columns of $Q$, equality holds if and only if \[ (H=1\text{ or }r=1)\qquad\text{and}\qquad(W=1\text{ or }c=1). \] Thus the minimum over all nonzero $s\times t$ patterns is $2^{HW}$, attained exactly by singleton patterns when $H,W>1$, and every fixed nonzero pattern has square growth rate $1$. We also refine the count by weight. If $o(Q)$ is the number of $1$-entries of $Q$, then the number of strongly $Q$-forcing matrices at the minimum positive weight $o(Q)$ is $\binom{H+r-1}{r}\binom{W+c-1}{c}$; at every fixed density in $(0,1)$, the logarithmic growth rate is the binary entropy when $m$ and $n$ are comparable. For ordinary forcing, where every $s\times t$ submatrix contains the $1$-entries of $Q$ in their prescribed positions, let $F(m,n,Q)$ be the number of forcing matrices and let $\mathfrak m(m,n,Q)$ be their minimum weight. We prove \[ F(m,n,Q)=2^{mn-\mathfrak m(m,n,Q)} \quad\text{and}\quad 2^{\,mn-\mathfrak m(m,n,Q)+HW} \le F(m,n,Q)F^{*}(m,n,Q) \le 2^{mn}. \] The lower product bound has the same equality cases as the strong-forcing lower bound above, while the upper product bound is attained exactly by singleton patterns. In particular, the product is at least $2$, with equality exactly when $s=m$, $t=n$, and $Q$ is the all-ones pattern.

We prove an asymptotic structural property of generic initial ideals. This single phenomenon yields results on Hilbert functions, persistence, hyperplane restriction, graded Betti numbers, combinatorial shadow minimization, and Lefschetz properties.

In this paper, we establish a new variant of the Burger-Brooks transfer principle, which allows us to apply spectral estimates for measured Riemannian surfaces to obtain the following result: There exists a universal constant $C>0$ such that, for every connected graph $G=(V, E)$ with boundary $B$, maximum degree $d_{\max}$ and genus $g$, \[\sigma_k(G, B)\leq C d_{\max}\frac{g+k}{|B|},\] where $1\leq k\leq |B|$ and $\sigma_k(G, B)$ denotes the $k$-th Steklov eigenvalue of $G$ with boundary $B$. This bound is sharp up to a universal constant, thereby resolving a problem raised by Lin and Zhao [J. Lond. Math. Soc. (2) 112 (2025), Paper No. e70238]. Furthermore, when $B=V$, the above result yields an upper bound for the Laplacian eigenvalues of graphs, improving the previously known bounds of Kelner, Lee, Price and Teng [Geom. Funct. Anal. 21 (2011), 1117–1143] and Amini and Cohen-Steiner [Comment. Math. Helv. 93 (2018), 203–223].

Polygon equations form a family of equations generalizing the pentagon equation. In this paper, we construct semi-simplicial sets of permitted colorings associated with set-theoretic solutions of polygon equations and use them to define the corresponding (co)homology groups. We investigate several properties of these groups and establish an equivalence of categories between set-theoretic solutions of polygon equations and higher Segal semi-simplicial sets satisfying certain conditions. As a special case, our result recovers the correspondence between bijective set-theoretic solutions of the pentagon equation and $2$-Segal semi-simplicial sets proved by Dyckerhoff–Kapranov.

Let $K_{r,s,t}$, with $r\le s\le t$, denote the complete tripartite graph whose partite sets have sizes $r,s,t$. Mahmoodian and Mirzakhani gave three necessary conditions for $K_{r,s,t}$ to admit a decomposition into 5-cycles and conjectured that these conditions are sufficient. One of the conditions is $t\le 4rs/(r+s)$. We prove the conjecture for every odd triple on the extremal boundary $t = 4rs/(r+s)$. The proof is constructive. After reducing an arbitrary odd boundary triple to $(r,s,t)=(hga,hgb,hab)$, $a+b=4g$, we give an explicit cyclic decomposition of $K_{ga,gb,ab}$ and use the Mahmoodian and Mirzakhani scaling theorem to supply the common factor $h$. Together with the previously known all-even result, this settles the conjecture for every triple satisfying the boundary condition with equality. We also report explicit computer-generated $C_5$-decompositions for 117 odd triples satisfying the necessary conditions, 116 of which are strict-interior cases. To the best of our knowledge, all 117 cases were previously unresolved: no decomposition for any of them had been reported, and none of the 117 triples is covered by earlier existence results, constructions, or their recursive consequences. Moreover, these 117 certificates together with the boundary construction settle every previously unresolved triple satisfying the necessary conditions with fewer than $4400$ edges. Each computation is supplied as a machine-readable cycle-list certificate and can be checked independently by a short Python verifier. We also give a complete human-readable edge-label-matrix certificate for $K_{9,19,23}$.

Let $\binom{[n]}{k}$ be the set of all $k$-element subsets of the set $\{1,\ldots,n\}$ and let $\mathcal A,\mathcal B \subseteq \binom{[n]}{k}$ be two cross-intersecting families, that is, $A\cap B\neq \emptyset$ for any $A\in \mathcal A$ and $B\in \mathcal B$. The classical cross-intersecting version of the Erdős–Ko–Rado theorem, due to Pyber and Matsumoto–Tokushige, states that if $n\geq 2k$, then $|\mathcal A||\mathcal B|\leq \binom{n-1}{k-1}^2,$ where the equality holds for $n>2k$ if and only if $\mathcal A=\mathcal B$ is a star. In the present paper, we first give a stability result of this theorem by using Filmus's FKN theorem on the slice and linear algebra method as follows: There exists a constant $C>1$ such that if $n\geq 2.07k$ and $|\mathcal A||\mathcal B|\geq (1-\epsilon)\binom{n-1}{k-1}^2$, where $\epsilon\leq \frac{k^2}{C^2 n ^2 }$, then there is a star $\mathcal{S}$ such that $|\mathcal{S} \Delta \mathcal A|\leq C \epsilon \binom{n}{k}$ and $|\mathcal{S} \Delta \mathcal B|\leq C \epsilon \binom{n}{k}.$ Moreover, based on this stability result and the eigenvalues of the matrices of the Johnson scheme, we present an Erdős–Ko–Rado theorem for cross-intersecting families in the Euclidean inner product showing that if $n\geq 2k$ and $k\geq d \geq 0$, then $$\big\langle\mathbf{v}_d(\mathcal A),\mathbf{v}_d(\mathcal B)\big\rangle \leq \frac{\binom{k}{d}\binom{k-1}{d}}{\binom{n-1}{d}}\binom{n-1}{k-1}^2 +\binom{k-1}{d-1} \binom{n-d-1}{k-d}\binom{n-1}{k-1},$$ together with uniqueness and a corresponding stability result, where $\mathbf{v}_d(\mathcal A) \in \mathbb R^{\binom{[n]}{d}}$ is the $d$-degree vector of $\mathcal A$ whose $U$-entry is the number of members in $\mathcal A$ containing $U$.

For a graph $G$ with $3\mid e(G)$, let $R(G,\mathbb Z_3)$ denote the least integer $N$ such that every edge weighting $w:E(K_N)\to\mathbb Z_3$ admits a copy of $G$ whose edge weights sum to zero. We prove that every $n$-vertex graph $G$ with $n\ge6$ and $3\mid e(G)$ satisfies $R(G,\mathbb Z_3)\le n+5$. Moreover, the stronger bound $R(G,\mathbb Z_3)\le n+4$ holds whenever $G$ contains an induced $P_4$ or an induced $2K_2$. Consequently, we resolve a problem posed by Caro and Mifsud: every graph $G$ with $3\mid e(G)$ satisfies $R(G,\mathbb Z_3)\le |V(G)|+8$, with equality if and only if $G\cong K_3$.

We exhibit two explicit circulant graphs of prime order $251$ that are $K_4$-free and have independence number $19$. Consequently \[R(4,20)\ge 252.\] These improve the bound $R(4,20)\ge 237$ given by Nagda, Raghavan, Thakurta and the long standing bound $R(4,21)\ge 242$ recorded in Radziszowski's dynamic survey. The graphs are $32$-subsets of a pair of undirected quintic cyclotomic classes modulo $251$, in analogy with the quartic-residue circulant of order $313$ used for $R(4,22)$. Clique-freeness is elementary; the independence-number claims are certified by a bitset branch-and-bound on the $186$-vertex residual of a vertex.

A graph $H$ is $K_{r+1}$-Turán-good if, for every sufficiently large $n$, the Turán graph $T_r(n)$ maximizes the number of copies of $H$ among all $n$-vertex $K_{r+1}$-free graphs. It is strictly $K_{r+1}$-Turán-good if $T_r(n)$ is the unique extremal graph. Morrison, Nir, Norin, Rzążewski and Wesolek [JCTB, 2023] proved that every graph $H$ is $K_{r+1}$-Turán-good whenever $r\ge 300v(H)^9$. They raised the following two questions: 1.Can the sufficient condition $r\ge 300v(H)^9$ be reduced to a condition of quadratic order in $v(H)$? 2.Is the Turán-good property monotone in $r$? More precisely, if a graph $H$ is $K_r$-Turán-good, must it also be $K_{r+1}$-Turán-good? We affirmatively resolve the first question and derive an even stronger bound linear in the edge number: every graph $H$ with at least one edge is strictly $K_{r+1}$-Turán-good and $K_{r+1}$-Turán-stable whenever $r\ge 168e(H)$. This condition is quadratic in $v(H)$ for arbitrary graphs and linear in $v(H)$ for every sparse graph family with $e(H)=O(v(H))$. We answer the second question negatively. For every $r\ge3$, there exists a graph that is strictly $K_r$-Turán-good but not $K_{r+1}$-Turán-good. More quantitatively, for every sufficiently large $h$, there exists a graph $H$ with $v(H)\le h$ and an integer $r=h-2\sqrt h+O(1)$ such that $H$ is strictly $K_r$-Turán-good but not $K_{r+1}$-Turán-good. The monotonicity threshold $\lambda(H)$ is the least integer $R\ge 2$ such that, for every $r\ge R$, the graph $H$ is $K_{r+1}$-Turán-good whenever it is $K_r$-Turán-good. For \[ \lambda_{\max}(h)=\max\{\lambda(H)\mid v(H)\le h\}, \] our two results yield \[ h-2\sqrt h-O(1)\le \lambda_{\max}(h)\le 84h^2. \]

We develop a unified theory of caged retractions of discrete polymatroids. Given a polymatroid and a cage $\kappa$, the $\kappa$-retraction is a canonical $\kappa$-caged polymatroid obtained by projecting bases into the cage and retaining the maximal projected bases. We prove that this construction agrees with an explicit rank-function formula. We show that the inclusion of the $\kappa$-caged polymatroids into all polymatroids and the $\kappa$-retraction form a Galois connection with respect to the weak-map order. As applications, we obtain caged versions of polymatroid union, the disjoint basis theorem, and induction along a bipartite graph. When $\kappa=\textbf{1}$, these recover the corresponding matroid constructions. We also study how caged retractions interact with Lorentzian polynomials and representations over near-idempotent tracts. In each case, the construction preserves the relevant structure.