Motivated by the construction of optimal locally repairable codes, we introduce the new finite geometric concept of a local arc which is defined as a collection S of disjoint point sets Si in PG(2,q) such that Si∪Sj is an arc for any Si,Sj∈S. We focus on the upper and lower bounds on the sizes of maximum k-uniform local arcs. For q=pm with p prime, we construct k-uniform local arcs in PG(2,q) of size Ω(qd) where d is between 1.1167 and 1.25 depending only on m. For k=4, this implies the existence of optimal locally repairable codes (LRCs) with minimum distance 6, locality 3, and disjoint repair groups, whose length is superlinear in q–a significant improvement over the previously known O(q) constructions for such LRCs.

Twin-width is a recently introduced graph parameter based on the repeated contraction of near-twins. It has shown remarkable utility in algorithmic and structural graph theory, as well as in finite model theory -- particularly since first-order model checking is fixed-parameter tractable when a witness certifying small twin-width is provided. However, the behavior of twin-width in specific graph classes, particularly cubic graphs, remains poorly understood. While cubic graphs are known to have unbounded twin-width, no explicit cubic graph of twin-width greater than 4 is known. This paper explores this phenomenon in regular and near-regular graph classes. We show that extremal graphs of bounded degree and high twin-width are asymmetric, partly explaining their elusiveness. Additionally, we establish bounds for circulant and d-degenerate graphs, and examine strongly regular graphs, which exhibit similar behavior to cubic graphs. Our results include determining the twin-width of Johnson graphs over 2-sets, and cyclic Latin square graphs.

The famous Erdos-Rado sunflower conjecture suggests that an s-sunflower-free family of k-element sets has size at most (C-s)(k) for some absolute constant C. In this note, we investigate the analog problem for k-spaces over the field with q elements. For s > k + 1, we show that the largest s-sunflower-free family T satisfies 1 <= |F|/q((s-1))(k+1 2 )(-k) < (q/(q-1))(k). For s < k, we show that q(-(k+1 2)) < |F|/q((s-1))(k+1 2 )-k <= (q/(q-1))(k). Our lower bounds rely on an iterative construction that uses lifted maximum rank-distance (MRD) codes.(c) 2025 Elsevier Inc. All rights are reserved, including those for text and datamining, AI training, and similar technologies

We give a bijection between the point-hyperplane antiflags of V(n, 2) and the nonsingular points of V(2n, 2) with respect to a hyperbolic quadric. With the help of this bijection, we give a description of the strongly regular graph NO^+_2n(2) in V(2n, 2). We also describe a graph with respect to a hyperbolic quadric in V(2n, 2) that was recently defined by Stanley and Takeda in V(n, 2). Similarly, we give a bijection between the point-hyperplane antiflags of V(n, 3) and the nonsingular points of one type in V(2n, 3) with respect to a hyperbolic quadric.

We provide an explicit algebraic construction showing that, uniformly for integers 3 ≤ s ≤ t, as t →∞, R( s,t ) ≥ t^(1-o(1)) log s / log(log s + 1) . For large fixed s, this improves the dependence on s in the general off-diagonal construction of Alon and Pudlák. In particular, R(33, t) ≥ t^2.1-o(1), to our knowledge, the first explicit construction showing R(s, t) ≥ t^c for some fixed s and some c > 2. In the diagonal case, it improves the leading constant in the exponent of the classical Frankl–Wilson bound from 1/4 to 1, while being almost as simple to describe.

Given a finite Lie incidence geometry which is either a polar space of rank at least $3$ or a strong parapolar space of symplectic rank at least $4$ and diameter at most $4$, or the parapolar space arising from the line Grassmannian of a projective space of dimension at least $4$, we show that its point graph is determined by its local structure. This follows from a more general result which classifies graphs whose local structure can vary over all local structures of the point graphs of the aforementioned geometries. In particular, this characterises the strongly regular graphs arising from the line Grassmannian of a finite projective space, from the half spin geometry related to the quadric $Q^+(10,q)$ and from the exceptional group of type $\mathsf{E_6}(q)$ by their local structure.

There are 6 families of finite polar spaces of rank $3$. The set of lines in a rank $3$ polar space form a rank $5$ association scheme. We determine the regular sets of minimal size in several of these polar spaces, and describe some examples. We also give a new family of Cameron--Liebler sets of generators in the polar spaces $O^+(10,q)$ when $q = 3^h$ using a regular set of lines in $O(7,q)$.

We construct a new family of distance-biregular graphs related to hyperovals and a new sporadic example of a distance-biregular graph related to Mathon's perp system. The infinite family can be explained using 2--homogeneity, while the sporadic example belongs to a generalization of a construction by Delorme. Additionally, we establish a new non-existence condition for distance-biregular graphs which, for instance, rules out the existence of a distance-biregular graph on 225+60 vertices.

A family ℱ of spanning trees of the complete graph on n vertices K_n is t-intersecting if any two members have a forest on t edges in common. We prove an Erdős–Ko–Rado result for t-intersecting families of spanning trees of K_n. In particular, we show there exists a constant C > 0 such that for all n ≥ C (log n) t the largest t-intersecting families are the families consisting of all trees that contain a fixed set of t disjoint edges (as well as the stars on n vertices for t = 1). The proof uses the spread approximation technique in conjunction with the Lopsided Lovász Local Lemma.

A minimum storage regenerating (MSR) subspace family of $\mathbb{F}_q^{2m}$ is a set $\mathcal{S}$ of $m$-spaces in $\mathbb{F}_q^{2m}$ such that for any $m$-space $S$ in $\mathcal{S}$ there exists an element in $\mathrm{PGL}(2m, q)$ which maps $S$ to a complement and fixes $\mathcal{S} \setminus \{ S \}$ pointwise. We show that an MSR subspace family of $2$-spaces in $\mathbb{F}_q^4$ has at most size $6$ with equality if and only if it is a particular subset of a Segre variety. This implies that an $(n, n-2, 4)$-MSR code has $n \leq 9$.

We construct a new family of strongly regular graphs with the same parameters as the strongly regular graphs $D_{5,5}(q)$. The construction can be seen as a variant of the construction of twisted Grassmann graphs by Van Dam and Koolen.

We classify the Boolean degree 1 functions of k-spaces in a vector space of dimension n (also known as Cameron-Liebler classes ) over the field with q elements for n >= n(0) (k, q). This also implies that two-intersecting sets with respect to k-spaces do not exist for n >= n(0) (k, q). Our main ingredient is the Ramsey theory for geometric lattices.

We survey results for Cameron-Liebler sets and low degree Boolean functions for Hamming graphs, Johnson graphs and Grassmann graphs from the point of view of association schemes. This survey covers selected results in finite geometry, Boolean function analysis, design theory, coding theory, and cryptography.

We use p-rank bounds on partial ovoids and the classical bounds on Ramsey numbers to obtain upper bounds on the size of partial m-ovoids in finite classical polar spaces. These bounds imply a uniform non-existence result of m-ovoids over all families of finite classical polar spaces. In the special case of the symplectic spaces over the binary field, we prove an equivalence between partial m-ovoids and a generalisation of Oddtown families from extremal set theory that has been studied under the name of m-nearly orthogonal sets. We give a new construction for large partial 2-ovoids in these spaces and thus 2-nearly orthogonal sets over the binary field. This construction uses triangle-free graphs associated to certain BCH codes whose complements have low 2-rank and it gives an asymptotic improvement over the previous best constructions. We give another construction of triangle-free graphs using a binary projective cap, which has low complementary rank over the reals. This improves the bounds in the recently introduced rank-Ramsey problem of Beniamini, Linial, and Shraibman. It also gives better constructions of large partial m-ovoids for m > 2 in the binary symplectic space.

We investigate what we call generalized ovoids, that is families of totally isotropic subspaces of finite classical polar spaces such that each maximal totally isotropic subspace contains precisely one member of that family. This is a generalization of ovoids in polar spaces as well as the natural q-analog of a subcube partition of the hypercube (which can be seen as a polar space with q=1 ). Our main result proves that a generalized ovoid of k-spaces in polar spaces of large rank does not exist.

Given positive integers e_1,e_2 , let X_i denote the set of e_i -dimensional subspaces of a fixed finite vector space V=(𝔽_q)^e_1+e_2 . Let Y_i be a non-empty subset of X_i and let α _i = |Y_i|/|X_i| . We give a positive lower bound, depending only on α _1,α _2,e_1,e_2,q , for the proportion of pairs (S_1,S_2)∈ Y_1× Y_2 which intersect trivially. As an application, we bound the proportion of pairs of non-degenerate subspaces of complementary dimensions in a finite classical space that intersect trivially. This problem is motivated by an algorithm for recognizing classical groups. By using techniques from algebraic graph theory, we are able to handle orthogonal groups over the field of order 2, a case which had eluded Niemeyer, Praeger, and the first author.

We survey the area of strongly regular graphs satisfying the 4-vertex condition and find several new families. We describe a switching operation on collinearity graphs of polar spaces that produces cospectral graphs. The obtained graphs satisfy the 4-vertex condition if the original graph belongs to a symplectic polar space.

For an incidence geometry 𝒢= (𝒫, ℒ, I) with a linear representation 𝒯_2^*(𝒦) , we apply WQH switching to construct a non-geometric graph Γ ' cospectral with the line graph Γ of 𝒢 . As an application, we show that for h ≥ 2 and 0< m < h , there are strongly regular graphs with parameters (v, k, λ , μ ) = (2^2 h (2^m+h+2^m-2^h), 2^h (2^h+1)(2^m-1), 2^h (2^m+1-3), 2^h (2^m-1)) which are not point graphs of partial geometries of order (s,t,α ) = ((2^h+1)(2^m-1), 2^h-1, 2^m-1) .

An affine vector space partition of AG(n, q) is a set of proper affine subspaces that partitions the set of points. Here we determine minimum sizes and enumerate equivalence classes of affine vector space partitions for small parameters. We also give parametric constructions for arbitrary field sizes.

We investigate the existence of Boolean degree $d$ functions on the Grassmann graph of $k$-spaces in the vector space $\mathbb{F}_q^n$. For $d=1$ several non-existence and classification results are known, and no non-trivial examples are known for $n \geq 5$. This paper focusses on providing a list of examples on the case $d=2$ in general dimension and in particular for $(n, k)=(6,3)$ and $(n,k) = (8, 4)$. We also discuss connections to the analysis of Boolean functions, regular sets/equitable bipartitions/perfect 2-colorings in graphs, $q$-analogs of designs, and permutation groups. In particular, this represents a natural generalization of Cameron-Liebler line classes.