Carlos Hoppen, Hanno Lefmann, Knut Odermann
In this note, we adapt the Keevash-Sudakov proof of the (Turán) Stability Theorem for the Fano plane to find an explicit dependency between the parameters $\varepsilon$ and $δ$. This is useful in the solution of a multicolored version for hypergraphs of an extremal problem about edge-colorings, known as the Erdős-Rothschild problem, which may be considered for the Fano plane.
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A Note on Fano Planes in Orthogonal Buekenhout-Metz Unitals of Even Order
该研究拓展了Feng和Li关于O'Nan构型的工作，将相关构造推广到偶阶正交Buekenhout-Metz酉平面的Fano平面中，发现了新的O'Nan构型并提出相关猜想。
This article gives an overview of toric Fano and toric weak Fano varieties associated to graphs and building sets. We also study some properties of such toric Fano varieties and discuss related topics.
It is shown that the automorphism group of a binary $q$-analog of the Fano plane is either trivial or of order $2$.
The $q$-Fano plane is the $q$-analog of the Steiner system $S(2,3,7)$. For any given prime power $q$ it is not known whether the $q$-Fano plane exists or not. We consider the structure of the $q$-Fano plane for any given $q$ and conclude that most of its structure is known. Even so, we were unable to determine whether it exists or not. A special attention is given for the case $q=2$ which was considered by most researchers before.
The Fano plane is the unique linear 3-uniform hypergraph on seven vertices and seven hyperedges. It was recently proved that, for all $n \geq 8$, the balanced complete bipartite 3-uniform hypergraph on $n$ vertices, denoted by $B_n$, is the 3-uniform hypergraph on $n$ vertices with the largest number of hyperedges that does not contain a copy of the Fano plane. For sufficiently large $r$ and $n$, we show that $B_n$ admits the largest number of $r$-edge colorings with no rainbow copy of the Fano plane.
One of the most intriguing problems, in $q$-analogs of designs, is the existence question of an infinite family of $q$-analog of Steiner systems, known also as $q$-Steiner systems, (spreads not included) in general, and the existence question for the $q$-analog of the Fano plane, known also as the $q$-Fano plane, in particular. These questions are in the front line of open problems in block design. There was a common belief and a conjecture that such structures do not exist. Only recently, $q$-Steiner systems were found for one set of parameters. In this paper, a definition for the $q$-analog …
A $q$-analogue of a $t$-design is a set $S$ of subspaces (of dimension $k$) of a finite vector space $V$ over a field of order $q$ such that each $t$ subspace is contained in a constant $λ$ number of elements of $S$. The smallest nontrivial feasible parameters occur when $V$ has dimension $7$, $t=2$, $q=2$, and $k=3$; which is the $q$-analogue of a $2$-$(7,3,1)$ design, the Fano plane. The existence of the binary $q$-analogue of the Fano plane has yet to be resolved, and it was shown by Kiermaier et al. (2016) that such a configuration must have an automorphism group of order at most $2$. We s…
Confirming a conjecture of Vera T. Sós in a very strong sense, we give a complete solution to Turán's hypergraph problem for the Fano plane. That is we prove for $n\ge 8$ that among all $3$-uniform hypergraphs on $n$ vertices not containing the Fano plane there is indeed exactly one whose number of edges is maximal, namely the balanced, complete, bipartite hypergraph. Moreover, for $n=7$ there is exactly one other extremal configuration with the same number of edges: the hypergraph arising from a clique of order $7$ by removing all five edges containing a fixed pair of vertices. For sufficient…
The smallest set of admissible parameters of a $q$-analog of a Steiner system is $S_2[2,3,7]$. The existence of such a Steiner system -- known as a binary $q$-analog of the Fano plane -- is still open. In this article, the automorphism group of a putative binary $q$-analog of the Fano plane is investigated by a combination of theoretical and computational methods. As a conclusion, it is either rigid or its automorphism group is cyclic of order $2$, $3$ or $4$. Up to conjugacy in $\operatorname{GL}(7,2)$, there remains a single possible group of order $2$ and $4$, respectively, and two possible…
An 'induced restriction' of a simple binary matroid $M$ is a restriction $M|F$, where $F$ is a flat of $M$. We consider the class $\mathcal{M}$ of all simple binary matroids $M$ containing neither a free matroid on three elements (which we call a 'claw'), nor a Fano plane as an induced restriction. We give an exact structure theorem for this class; two of its consequences are that the matroids in $\mathcal{M}$ have unbounded critical number, while the matroids in $\mathcal{M}$ not containing the clique $M(K_5)$ as an induced restriction have critical number at most $2$.
Perfect matchings, Fano planes, and orthogonal bases of type $E_8$
利用完美匹配和标记Fano平面构造并研究$E_8$根系中的2025个正交基，这些基构成一个Bruhat型偏序集，通过广义Rothe图的基数计算秩函数，并给出组合刻画与秩生成函数。
The hypergraph Ramsey number of two $3$-uniform hypergraphs $G$ and $H$, denoted by $R(G,H)$, is the least integer $N$ such that every red-blue edge-coloring of the complete $3$-uniform hypergraph on $N$ vertices contains a red copy of $G$ or a blue copy of $H$. The Fano plane $\mathbb{F}$ is the unique 3-uniform hypergraph with seven edges on seven vertices in which every pair of vertices is contained in a unique edge. There is a simple construction showing that $R(H,\mathbb{F}) \ge 2(v(H)-1) + 1.$ Hypergraphs $H$ for which the equality holds are called $\mathbb{F}$-good. Conlon asked to dete…
Given a 3-graph H, let \ex_2(n, H) denote the maximum value of the minimum codegree of a 3-graph on n vertices which does not contain a copy of H. Let F denote the Fano plane, which is the 3-graph \{axx',ayy',azz',xyz',xy'z,x'yz,x'y'z'\}. Mubayi proved that \ex_2(n,F)=(1/2+o(1))n and conjectured that \ex_2(n, F)=\floor{n/2} for sufficiently large n. Using a very sophisticated quasi-randomness argument, Keevash proved Mubayi's conjecture. Here we give a simple proof of Mubayi's conjecture by using a class of 3-graphs that we call rings. We also determine the Turán density of the family of rings…
A conjecture widely attributed to Neumann is that all finite non-desarguesian projective planes contain a Fano subplane. In this note, we show that any finite projective plane of even order which admits an orthogonal polarity contains a Fano subplane. The number of planes of order less than $n$ previously known to contain a Fano subplane was $O(\log n)$, whereas the number of planes of order less than $n$ that our theorem applies to is not bounded above by any polynomial in $n$.
This paper classifies toric Fano 3-folds with singular locus { 1/k(1,1,1) } for any positive integer k, building on the work of Batyrev and Watanabe-Watanabe. This is achieved by completing an equivalent problem in the language of Fano polytopes. Furthermore we identify birational relationships between entries of the classification. For a fixed value k>4, there are exactly two such Fano 3-folds linked by a blow-up in a torus-invariant line.
Given a seven-element set $X = \{1,2,3,4,5,6,7\}$, there are 30 ways to define a Fano plane on it. Let us call a line of such Fano plane, that is to say an unordered triple from $X$, ordinary or defective according as the sum of two smaller integers from the triple is or is not equal to the remaining one, respectively. A point of the labeled Fano plane is said to be of order $s$, $0 \leq s \leq 3$, if there are $s$ {\it defective} lines passing through it. With such structural refinement in mind, the 30 Fano planes are shown to fall into eight distinct types. Out of the total of 35 lines, nine…
In this article we consider structures on a Fano plane ${\cal F}$ which allow a generalisation of Freudenthal's construction of a norm and a bilinear multiplication law on an eight-dimensional vector space ${\mathbb O_{\cal F}}$ canonically associated to ${\cal F}$. We first determine necessary and sufficient conditions in terms of the incidence geometry of ${\cal F}$ for these structures to give rise to division composition algebras, and classify the corresponding structures using a logarithmic version of the multiplication. We then show how these results can be used to deduce analogous resul…
Orthogonal and oriented Fano planes, triangular embeddings of $K_7,$ and geometrical representations of the Frobenius group $F_{21}$
本文研究21阶Frobenius群F21的几何表示，证明两个正交Fano平面的公共自同构群与适当定向Fano平面的自同构群均同构于F21，并证明K7的任意三角嵌入同构于经典环面双嵌入且面2-可着色，从而给出F21的三种几何表示。
On $α$-points of $q$-analogs of the Fano plane
该研究围绕Fano平面$q$-模拟的存在性这一开放问题，探究其$α$-点性质，证明全$α$-点超平面蕴含辛广义四边形$W(q)$的展形划分，将已有非$α$-点阻塞集结论推广到所有素数及偶数$q$。
On putative q-Analogues of the Fano Plane and Related Combinatorial Structures
针对Fano平面q-模拟的存在性难题，研究者利用子空间码理论尝试构造，虽未成功，但得到了对应参数下目前已知最大的子空间码，包括创纪录的6977规模三元码和329规模的无计算机构造二元码。
Let $F$ be a field. A $2$-$(7, 3, 1)_F$-subspace design, or $q$-Fano plane, over $F$, is a $7$-dimensional vector space $V$ over $F$ together with a collection $\mathfrak{B}$ of three-dimensional subspaces of $V$ such that every two-dimensional subspace of $V$ is contained in exactly one element $B$ of $\mathfrak{B}$. The question of existence of $q$-Fano planes over any field has been open since the 1970s and has attracted considerable attention in the special case that $F$ is finite. Here we show the existence of $2$-$(7, 3, 1)_F$-subspace designs over certain infinite fields $F$, including …
In this note we report on examples of 7- and 8-dimensional toric Fano manifolds that are not symmetric and still admit a Kaehler-Einstein metric. This answers a question first posed by V.V. Batyrev and E. Selivanova. The examples were found in the classification of toric Fano manifolds up to dimension 8 obtained by M. Oebro. We also discuss related open questions and conjectures.
We show that there is a binary subspace code of constant dimension 3 in ambient dimension 7, having minimum distance 4 and cardinality 333, i.e., $333 \le A_2(7,4;3)$, which improves the previous best known lower bound of 329. Moreover, if a code with these parameters has at least 333 elements, its automorphism group is in one of $31$ conjugacy classes. This is achieved by a more general technique for an exhaustive search in a finite group that does not depend on the enumeration of all subgroups.
If parallel lines could meet: What exactly can a poet say about the Fano plane?
本文通过射影几何创作了新的诗歌形式，探讨数学与诗歌中的创造力互动，并提出未来研究方向。
Random Turán Theorem for the Fano Plane
该研究确定了随机3-均匀超图中，最大无F子超图是否为二分图的尖锐阈值，这是随机超图Turán型问题的首个尖锐阈值结果。
Exact Turán number of the Fano plane in the $\ell_2$-norm
本文证明了 Fano 平面在 $\ell_2$-范数下的 Andrásfai-Erdős-Sós 型稳定性定理，确认平衡完全二部 $3$-图为其唯一极值构造，解决了 Balogh-Clemen-Lidický 的猜想。
We continue our study and classification of structures on the Fano plane ${\cal F}$ and its dual ${\cal F}^\ast$ involved in the construction of octonions and the Lie algebra $\mathfrak g_2 (\mathbb F)$ over a field $\mathbb F$. These are a "composition factor": ${\cal F}\times {\cal F} \to\{-1, 1\}$, inducing an octonion multiplication, and a function $δ^\ast : Aut({\cal F}) \times {\cal F}^\ast \to \{-1, 1\}$ such that $g \in Aut({\cal F})$ can be lifted to an automorphism of the octonions iff $δ^\ast(g, \cdot)$ is the Radon transform of a function on ${\cal F}$. We lift the action of $Aut({…
Let $θ$ be an automorphism of a thick irreducible spherical building $Δ$ of rank at least $3$ with no Fano plane residues. We prove that if there exist both type $J_1$ and $J_2$ simplices of $Δ$ mapped onto opposite simplices by $θ$, then there exists a type $J_1\cup J_2$ simplex of $Δ$ mapped onto an opposite simplex by $θ$. This property is called "cappedness". We give applications of cappedness to opposition diagrams, domesticity, and the calculation of displacement in spherical buildings. In a companion piece to this paper we study the thick irreducible spherical buildings containing Fano …
Given the facts that the three-qubit symplectic polar space features three different kinds of observables and each of its labeled Fano planes acquires a definite sign, we found that there are 45 distinct types of Mermin pentagrams in this space. A key element of our classification is the fact that any context of such pentagram is associated with a unique (positive or negative) Fano plane. Several intriguing relations between the character of pentagrams' three-qubit observables and `valuedness' of associated Fano planes are pointed out. In particular, we find two distinct kinds of negative cont…
It has been shown by Hacking and Prokhorov that if the projective surface X with quotient singularities and self-intersection number 9 has a smoothing to the projective plane, then X is the general fiber of a Q-Gorenstein deformation of the weighted projective plane with weights giving solutions to the Markov equation. This result has been understood and generalized by combinatorial mutations of Fano triangles by Akhtar, Coates, Galkin, and Kasprzyk. In this note, we study this result by utilizing polarized T-varieties and describe the associated deformation explicitly in terms of certain Mink…
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