Largest set of unitals in projective planes of order 16

Unitals in $PG(2,q^2)$ with a large 2-point stabiliser

Let $\cU$ be a unital embedded in the Desarguesian projective plane $\PG(2,q^2)$. Write $M$ for the subgroup of $\PGL(3,q^2)$ which preserves $\cU$. We show that $\cU$ is classical if and only if $\cU$ has two distinct points $P,Q$ for which the stabiliser $G=M_{P,Q}$ has order $q^2-1$.

The largest projective cube-free subsets of $\mathbb{Z}_{2^n}$

In the Boolean lattice, Sperner's, Erdős's, Kleitman's and Samotij's theorems state that families that do not contain many chains must have a very specific layered structure. We show that if instead of $\mathbb{Z}_2^n$ we work in $\mathbb{Z}_{2^n}$, several analogous statements hold if one replaces the word $k$-chain by projective cube of dimension $2^{k-1}$. We say that $B_d$ is a projective cube of dimension $d$ if there are numbers $a_1, a_2, \ldots, a_d$ such that $$B_d = \left\{\sum_{i\in I} a_i \bigg\rvert \emptyset \neq I\subseteq [d]\right\}.$$ As an analog of Sperner's and Erdős's the…

Hilton-Milner results in projective and affine spaces

In this article, we analyse maximal sets of $k$-spaces, in PG(n,q) and AG(n,q), $n>2k+t+2$, that pairwise meet in at least a $t$-space. It is known that for both PG(n,q) and AG(n,q), the largest example is a $t$-pencil, i.e. the set of all $k$-spaces containing a fixed $t$-space. In this paper, we analyse the structure of the second largest maximal example in both PG(n,q) and AG(n,q).

The chromatic number of a two families of generalized Kneser graphs related to finite generalized quadrangles and finite projective 3-spaces

Let $Γ$ be the graph whose vertices are the chambers of the finite projective space $PG(3,q)$ with two vertices being adjacent when the corresponding chambers are in general position. It is known that the independence number of this graph is $(q^2+q+1)(q+1)^2$. For $q\ge 43$ we determine the largest independent set of $Γ$ and show that every maximal independent set that is not a largest one has at most constant times $q^3$ elements. For $q\ge 47$, this information is then used to show that $Γ$ has chromatic number $q^2+q$. Furthermore, for many families of generalized quadrangles we prove simi…

A Problem Concerning Nonincident Points and Lines in Projective Planes

In this paper, we study the problem of finding the largest possible set of s points and s lines in a projective plane of order q, such that that none of the s points lie on any of the s lines. We prove that s <= 1+(q+1)(\sqrt{q}-1). We also show that equality can be attained in this bound whenever q is an even power of two.

Largest regular multigraphs with three distinct eigenvalues

We deal with connected $k$-regular multigraphs of order $n$ that has only three distinct eigenvalues. In this paper, we study the largest possible number of vertices of such a graph for given $k$. For $k=2,3,7$, the Moore graphs are largest. For $k\ne 2,3,7,57$, we show an upper bound $n\leq k^2-k+1$, with equality if and only if there exists a finite projective plane of order $k-1$ that admits a polarity.

On the number of $k$-gons in finite projective planes

Let $Π$ be a projective plane of order $n$ and $Γ_Π$ be its Levi graph (the point-line incidence graph). For fixed $k \geq 3$, let $c_{2k}(Γ_Π)$ denote the number of $2k$-cycles in $Γ_Π$. In this paper we show that $$ c_{2k}(Γ_Π) = \frac{1}{2k}n^{2k} + O(n^{2k-2}), \hspace{0.5cm} n \rightarrow \infty. $$ We also state a conjecture regarding the third and fourth largest terms in the asymptotic of the number of $2k$-cycles in $Γ_Π$. This result was also obtained independently by Voropaev in 2012. Let $\text{ex}(v, C_{2k}, \mathcal{C}_{\text{odd}}\cup \{C_4\})$ denote the greatest number of $2k$-…

关于 PG(4,q) 的房间构成的 Kneser 图中最大独立集的研究

On the largest independent sets in the Kneser graph on chambers of PG(4,q)

本文研究了有限射影空间 PG(4,q) 房间构成的 Kneser 图 $Γ_4$，证明了当 $q\geq 749$ 时其独立数为 $α:=(q^2+q+1)(q^3+2q^2+q+1)(q+1)^2$，并刻画了最大独立集的几何结构。

Equivalence and automorphism groups of two families of maximum scattered linear sets

Linear set in projective spaces over finite fields plays central roles in the study of blocking sets, semifields, rank-metric codes and etc. A linear set with the largest possible cardinality and the maximum rank is called maximum scattered. Despite two decades of study, there are only a few number of known maximum scattered linear sets in projective lines, including the family constructed by Csajbók, Marino, Polverino and Zanella 2018, and the family constructed by Csajbók, Marino, Zullo 2018 (also Marino, Montanucci, and Zullo 2020). This paper aims to solve the equivalence problem of the li…

利用Grassmann图恢复底层射影几何

Using a Grassmann graph to recover the underlying projective geometry

本文利用Grassmann图对应于邻接矩阵第二大特征值的Euclidean表示，仅通过图结构恢复底层射影几何中两个顶点的交与和。

关于有限射影空间中的4-一般集

On 4-general sets in finite projective spaces

本文研究了有限射影空间 ${\rm PG}(n,q)$ 中完全 $4$-一般集大小的上下界，给出了 $q \in \{3,4\}$ 时接近理论上界的集合，并构造了特定条件下大小为 $3(q+1)$ 的传递 $4$-一般集。

关于类似于八元数射影平面的流形的27顶点三角剖分的可能对称群

On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane

本文研究了八元数射影平面流形的27顶点三角剖分的对称群限制，利用Smith和Bredon的不动点集拓扑结果，确定了S_27的26个可能子群，其中最大群阶为351，其余群阶均不超过52。