The celebrated Erdős–Ko–Rado (EKR) theorem for Paley graphs of square order states that all maximum cliques are canonical in the sense that each maximum clique arises from the subfield construction. Recently, Asgarli and Yip extended this result to Peisert graphs and other Cayley graphs which are Peisert-type graphs with nice algebraic properties on the connection set. On the other hand, there are Peisert-type graphs for which the EKR theorem fails to hold. In this article, we show that the EKR theorem of Paley graphs extends to almost all pseudo-Paley graphs of Peisert-type. Furthermore, we establish the stability results of the same flavor.

In set theory without the Axiom of Choice ( $\mathsf {AC}$ ), we investigate the open problem of the deductive strength of statements which concern the existence of almost disjoint and maximal almost disjoint (MAD) families of infinite-dimensional subspaces of a given infinite-dimensional vector space, as well as the extension of almost disjoint families in infinite-dimensional vector spaces to MAD families.

This work focuses on the ongoing research of lineability (the search for large linear structures within certain non-linear sets) in non-Archimedean frameworks. Among several other results, we show that there exist large linear structures inside each of the following sets: (i) functions with a fixed closed subset of continuity, (ii) all continuous functions that are not Darboux continuous (or vice versa), (iii) all functions whose Dieudonné integral does not behave as an antiderivative, and (iv) functions with finite range and having antiderivative.

We present a sufficient and necessary condition for a function module space $X$ to have the approximate hyperplane series property (AHSP). As a consequence, we have that the space ${\mathcal{C}}_{0}(L,E)$ of bounded and continuous $E$-valued mappings defined on the locally compact Hausdorff space $L$ has AHSP if and only if $E$ has AHSP.

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame, which encompasses families of arrangements. The frame uses the notion of matroids. For the proof of the existence of the potentials, a power series ansatz is made. The proof that it works requires that certain decompositions of tuples of coordinate vector fields are related by certain elementary transformations. This is shown with a nontrivial result on matroid partition.

We say that a function $h\,:\,\mathbb{R}\,\to \,\mathbb{R}$ is a Hamel function $(h\,\in \,\text{HF)}$ if $h$, considered as a subset of ${{\mathbb{R}}^{2}},$ is a Hamel basis for ${{\mathbb{R}}^{2}}.$ We show that $\text{A}\left( \text{HF} \right)\,\ge \,\omega$, i.e., for every finite $F\,\subseteq \,{{\mathbb{R}}^{\mathbb{R}}}$ there exists $f\,\in \,{{\mathbb{R}}^{\mathbb{R}}}$ such that $f\,+\,F\,\subseteq \,\text{HF}$. From the previous work of the author it then follows that $\text{A}\left( \text{HF} \right)\,=\,\omega$.

Hua’s fundamental theorem of the geometry of hermitian matrices characterizes bijective maps on the space of all $n\times n$ hermitianmatrices preserving adjacency in both directions. The problem of possible improvements has been open for a while. There are three natural problems here. Do we need the bijectivity assumption? Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only? Can we obtain such a characterization formaps acting between the spaces of hermitian matrices of different sizes? We answer all three questions for the complex hermitian matrices, thus obtaining the optimal structural result for adjacency preserving maps on hermitian matrices over the complex field.

The structure of Schur multiplicative maps on matrices over a field is studied. The result is then used to characterize Schur multiplicative maps f satisfying for different subsets S of matrices including the set of rank k matrices, the set of singular matrices, and the set of invertible matrices. Characterizations are also obtained for maps on matrices such that Γ(f(A))=Γ(A) for various functions Γ including the rank function, the determinant function, and the elementary symmetric functions of the eigenvalues. These results include analogs of the theorems of Frobenius and Dieudonné on linear maps preserving the determinant functions and linear maps preserving the set of singular matrices, respectively.

It is proved that every adjacency preserving continuous map on the vector space of real matrices of fixed size, is either a bijective affine tranformation of the form $A\,\mapsto \,PAQ\,+\,R$, possibly followed by the transposition if the matrices are of square size, or its range is contained in a linear subspace consisting of matrices of rank at most one translated by some matrix $R$. The result extends previously known theorems where the map was assumed to be also injective.

We investigate categorical and amalgamation properties of the functor $\operatorname{Id_c}$ assigning to every partially ordered abelian group $G$ its $\langle \vee, 0 \rangle$-semilattice of compact ideals $\operatorname{Id_c} G$. Our main result is the following.

Theorem 1. Every diagram of finite Boolean semilattices indexed by a finite dismantlable partially ordered set can be lifted, with respect to the$\operatorname{Id_c}$functor, by a diagram of pseudo-simplicial vector spaces.

Pseudo-simplicial vector spaces are a special kind of finite-dimensional partially ordered vector spaces with interpolation over the field of rational numbers. The methods introduced also make it possible to prove the following ring-theoretical result.

Theorem 2. For any countable distributive$\langle \vee, 0 \rangle$-semilattices$S$and$T$and any field$K$, any$\langle \vee, 0 \rangle$-homomorphism$f \colon S \to T$can be lifted, with respect to the$\operatorname{Id_c}$functor on rings, by a homomorphism $f \colon A \to B$of$K$-algebras, for countably dimensional locally matricial algebras$A$and$B$over$K$.

We also state a lattice-theoretical analogue of Theorem 2 (with respect to the $\operatorname{Con_c}$ functor), and we provide counterexamples to various related statements. In particular, we prove that the result of Theorem 1 cannot be achieved with simplicial vector spaces alone.