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Ferdinand Ihringer

Publications and source records attributed to Ferdinand Ihringer.

At least 55 records · Page 3Linked to original sources
arXiv2020

Remarks on the Erdős Matching Conjecture for Vector Spaces

In 1965, Paul Erdős asked about the largest family $Y$ of $k$-sets in $\{ 1, \ldots, n \}$ such that $Y$ does not contain $s+1$ pairwise disjoint sets. This problem is commonly known as the Erdős Matching Conjecture. We investigate the $q$-analog of this question, that is we want to determine the size of a largest family $Y$ of $k$-spaces in $\mathbb{F}_q^n$ such that $Y$ does not contain $s+1$ pairwise disjoint $k$-spaces. Here we call two subspaces disjoint if they intersect trivially. Our main result is, slightly simplified, that if $16 s \leq \min\{ q^{\frac{n-k}{4}},$ $q^{\frac{n-2k+1}{3}} \}$, then $Y$ is either small or a union of intersecting families. Thus we show the Erd\H{os} Matching Conjecture for this range. The proof uses a method due to Metsch. We also discuss constructions. In particular, we show that for larger $s$, there are large examples which are close in size to a union of intersecting families, but structurally different. As an application, we discuss the close relationship between the Erdős Matching Conjecture for vector spaces and Cameron-Liebler line classes (and their generalization to $k$-spaces), a popular topic in finite geometry for the last 30 years. More specifically, we propose the Erdős Matching Conjecture (for vector spaces) as an interesting variation of the classical research on Cameron-Liebler line classes.

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arXiv2020

New bounds on the Ramsey number $r(I_m, L_n)$

We investigate the Ramsey numbers $r(I_m, L_n)$ which is the minimal natural number $k$ such that every oriented graph on $k$ vertices contains either an independent set of size $m$ or a transitive tournament on $n$ vertices. Apart from the finitary combinatorial interest, these Ramsey numbers are of interest to set theorists since it is known that $r(ωm, n) = ωr(I_m, L_n)$, where $ω$ is the lowest transfinite ordinal number, and $r(κm, n) = κr(I_m, L_n)$ for all initial ordinals $κ$. Continuing the research by Bermond from 1974 who did show $r(I_3, L_3) = 9$, we prove $r(I_4, L_3) = 15$ and $r(I_5, L_3) = 23$. The upper bounds for both the estimates above are obtained by improving the upper bound of $m^2$ on $r(I_m, L_3)$ due to Larson and Mitchell (1997) to $m^2 - m + 3$. Additionally, we provide asymptotic upper bounds on $r(I_m, L_n)$ for all $n \geq 3$. In particular, we show that $r(I_m, L_3) \in Θ(m^2 / \log m)$.

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arXiv2020

New Bounds for Partial Spreads of $H(2d-1, q^2)$ and Partial Ovoids of the Ree-Tits Octagon

Two results are obtained that give upper bounds on partial spreads and partial ovoids respectively. The first result is that the size of a partial spread of the Hermitian polar space $\mathsf{H}(3, q^2)$ is at most $\left(\frac{2p^3+p}{3} \right)^t+1$, where $q=p^t$, $p$ is a prime. For fixed $p$ this bound is in $o(q^3)$, which is asymptotically better than the previous best known bound of $(q^3+q+2)/2$. Similar bounds for partial spreads of $\mathsf{H}(2d-1, q^2)$, $d$ even, are given. The second result is that the size of a partial ovoid of the Ree-Tits octagon $\mathsf{O}(2^t)$ is at most $26^t+1$. This bound, in particular, shows that the Ree-Tits octagon $\mathsf{O}(2^t)$ does not have an ovoid.

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arXiv2020

The Chromatic Number of the $q$-Kneser Graph for $q \geq 5$

We obtain a new weak Hilton-Milner type result for intersecting families of $k$-spaces in $\mathbb{F}_q^{2k}$, which improves several known results. In particular the chromatic number of the $q$-Kneser graph $qK_{n:k}$ was previously known for $n > 2k$ (except for $n=2k+1$ and $q=2$) or $k < q \log q - q$. Our result determines the chromatic number of $qK_{2k:k}$ for $q \geq 5$, so that the only remaining open cases are $(n, k) = (2k, k)$ with $q \in \{ 2, 3, 4 \}$ and $(n, k) = (2k+1, k)$ with $q = 2$.

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arXiv2020

A construction for clique-free pseudorandom graphs

A construction of Alon and Krivelevich gives highly pseudorandom $K_k$-free graphs on $n$ vertices with edge density equal to $Θ(n^{-1/(k -2)})$. In this short note we improve their result by constructing an infinite family of highly pseudorandom $K_k$-free graphs with a higher edge density of $Θ(n^{-1/(k - 1)})$.

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arXiv2019

On the Algebraic Combinatorics of Injections and its Applications to Injection Codes

We consider the algebraic combinatorics of the set of injections from a $k$-element set to an $n$-element set. In particular, we give a new combinatorial formula for the spherical functions of the Gelfand pair $(S_k \times S_n, \text{diag}(S_k) \times S_{n-k})$. We use this combinatorial formula to give new Delsarte linear programming bounds on the size of codes over injections.

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arXiv2019

The Independence Number of the Orthogonality Graph in Dimension $2^k$

We determine the independence number of the orthogonality graph on $2^k$-dimensional hypercubes. This answers a question by Galliard from 2001 which is motivated by a problem in quantum information theory. Our method is a modification of a rank argument due to Frankl who showed the analogous result for $4p^k$-dimensional hypercubes, where $p$ is an odd prime.

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arXiv2019

New Strongly Regular Graphs from Finite Geometries via Switching

We show that the strongly regular graph on non-isotropic points of one type of the polar spaces of type $U(n, 2)$, $O(n, 3)$, $O(n, 5)$, $O^+(n, 3)$, and $O^-(n, 3)$ are not determined by its parameters for $n \geq 6$. We prove this by using a variation of Godsil-McKay switching recently described by Wang, Qiu, and Hu. This also results in a new, shorter proof of a previous result of the first author which showed that the collinearity graph of a polar space is not determined by its spectrum. The same switching gives a linear algebra explanation for the construction of a large number of non-isomorphic designs.

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arXiv2019

Regular Intersecting Families

We call a family of sets intersecting, if any two sets in the family intersect. In this paper we investigate intersecting families $\mathcal{F}$ of $k$-element subsets of $[n]:=\{1,\ldots, n\},$ such that every element of $[n]$ lies in the same (or approximately the same) number of members of $\mathcal{F}$. In particular, we show that we can guarantee $|\mathcal{F}| = o({n-1\choose k-1})$ if and only if $k=o(n)$.

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arXiv2018

The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters

We prove a conjecture by Van Dam and Sotirov on the smallest eigenvalue of (distance-$j$) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-$j$) Johnson graphs. More generally, we study the smallest eigenvalue and the second largest eigenvalue in absolute value of the graphs of the relations of classical $P$- and $Q$-polynomial association schemes.

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arXiv2018

Boolean constant degree functions on the slice are juntas

We show that a Boolean degree $d$ function on the slice $\binom{[n]}{k} = \{ (x_1,\ldots,x_n) \in \{0,1\} : \sum_{i=1}^n x_i = k \}$ is a junta, assuming that $k,n-k$ are large enough. This generalizes a classical result of Nisan and Szegedy on the hypercube. Moreover, we show that the maximum number of coordinates that a Boolean degree $d$ function can depend on is the same on the slice and the hypercube.

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arXiv2017

The binary $q$-analogue of the Fano plane has a trivial automorphism group

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 show that the binary $q$-analogue of the Fano plane has a trivial automorphism group.

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arXiv2017

Manickam-Miklós-Singhi Conjectures on Partial Geometries

In this paper we give a proof of the Manickam-Miklós-Singhi (MMS) conjecture for some partial geometries. Specifically, we give a condition on partial geometries which implies that the MMS conjecture holds. Further, several specific partial geometries that are counter-examples to the conjecture are described.

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arXiv2017

A Switching for all Strongly Regular Collinearity Graphs From Polar Spaces

We describe a general construction of strongly regular graphs from the collinearity graph of a finite classical polar spaces of rank at least $3$ over a finite field of order $q$. We show that these graphs are non-isomorphic to the collinearity graphs and have the same parameters. To our knowledge for most of these parameters these graphs are new as the collinearity graphs were the only known examples.

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arXiv2016

Infection in Hypergraphs

In this paper a new parameter for hypergraphs called hypergraph infection is defined. This concept generalizes zero forcing in graphs to hypergraphs. The exact value of the infection number of complete and complete bipartite hypergraphs is determined. A formula for the infection number for interval hypergraphs and several families of cyclic hypergraphs is given. The value of the infection number for a hypergraph whose edges form a symmetric t-design is given, and bounds are determined for a hypergraph whose edges are a t-design. Finally, the infection number for several hypergraph products and line graphs are considered.

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arXiv2015

Large $\{0, 1, \ldots, t\}$-Cliques in Dual Polar Graphs

We investigate $\{0, 1, \ldots, t \}$-cliques of generators on dual polar graphs of finite classical polar spaces of rank $d$. These cliques are also known as Erdős-Ko-Rado sets in polar spaces of generators with pairwise intersections in at most codimension $t$. Our main result is that we classify all such cliques of maximum size for $t \leq \sqrt{8d/5}-2$ if $q \geq 3$, and $t \leq \sqrt{8d/9}-2$ if $q = 2$. We have the following byproducts. (a) For $q \geq 3$ we provide estimates of Hoffman's bound on these $\{0, 1, \ldots, t \}$-cliques for all $t$. (b) For $q \geq 3$ we determine the largest, second largest, and smallest eigenvalue of the graphs which have the generators of a polar space as vertices and where two generators are adjacent if and only if they meet in codimension at least $t+1$. Furthermore, we provide nice explicit formulas for all eigenvalues of these graphs. (c) We provide upper bounds on the size of the second largest maximal $\{0, 1, \ldots, t \}$-cliques for some $t$.

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arXiv2015

Weighted Intriguing Sets in Finite Polar Spaces

We provide new proofs for the non-existence of ovoids in hyperbolic spaces of rank at least four in even characteristic, and for the Hermitian polar space $\mathsf{H}(5, 4)$. We also improve the results of A. Klein on the non-existence of ovoids of Hermitian spaces and hyperbolic quadrics.

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