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Ferdinand Ihringer

Publications and source records attributed to Ferdinand Ihringer.

At least 19 recordsLinked to original sources
arXiv2026

An improved algebraic construction for Ramsey numbers

We provide an explicit algebraic construction showing that, uniformly for integers $3 \leq s \leq t$, as $t \to \infty$, \[ R( s,t ) \geq t^{(1-o(1)) \log s / \log(\log s + 1) }. \] For large fixed $s$, this improves the dependence on $s$ in the general off-diagonal construction of Alon and Pudl\'ak. In particular, $R(33, t) \geq t^{2.1-o(1)}$, to our knowledge, the first explicit construction showing $R(s, t) \geq t^c$ 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.

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arXiv2026

Structure of $t$-Intersecting Families of Vector Spaces

We study $t$-intersecting and $t$-cross-intersecting families of $k$-dimensional subspaces in finite vector spaces of dimension $n$. We show that all large $t$-intersecting families admit a governing low-dimensional structure for $n \ge 2k+1$. This result, together with its cross-intersecting variant, allows us to prove analogues of several classical extremal set-theoretic results. In particular, we determine the intersecting families with the largest diversity, and we establish a Frankl-type degree-diversity result that generalizes the Hilton-Milner theorem. Our proofs rely on simplification procedures for $t$-intersecting and $t$-cross-intersecting families of subspaces. These procedures are based on the concept of subspace spreadness, a generalization of the classical notion of spreadness for set systems.

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arXiv2026

On the construction of large local arcs

Motivated by the construction of optimal locally repairable codes, we introduce the new finite geometric concept of a \emph{local arc} which is defined as a collection $\mathcal{S}$ of disjoint point sets $S_{i}$ in $\mathrm{PG}(2,q)$ such that $S_{i} \cup S_{j}$ is an arc for any $S_{i}, S_{j} \in \mathcal{S}$. We focus on the upper and lower bounds on the sizes of maximum $k$-uniform local arcs. For $q=p^m$ with $p$ prime, we construct $k$-uniform local arcs in $\mathrm{PG}(2,q)$ of size $\Omega(q^{d})$ 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.

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arXiv2026

Skirting the $n$-tuples

Let $n\ge 2$ and $q\ge 2$ be given. The set $X = \mathbb Z_q^n$ is a metric space of diameter $n$ under the Hamming metric $d(\cdot,\cdot)$. We seek a smallest set $S\subseteq X$ that ``skirts'' every $q$-ary $n$-tuple in the sense that every $x\in X$ is at distance $n$ from at least one element of $S$. Thus we aim to compute the total domination number $f(n,q)$ of the graph $G(n,q)$ with vertex set $X$ and edge set $\{ xy \, \| \, d(x,y)=n\}$. We provide constructions and bounds for this number, establishing $f(n,q) = C_q^{(1+o(1))n}$ for some constants $2=C_2>C_3 \geq \cdots$ which we are only able to estimate at the present time.

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arXiv2025

The Erdős-Rado Sunflower Problem for Vector Spaces

The famous Erdős-Rado sunflower conjecture suggests that an $s$-sun\-flower-free family of $k$-element sets has size at most $(Cs)^k$ for some absolute constant $C$. In this note, we investigate the analog problem for $k$-spaces over the field with $q$ elements. For $s \geq k+1$, we show that the largest $s$-sunflower-free family $\mathcal{F}$ satisfies \[ 1 \leq |\mathcal{F}| / q^{(s-1) \binom{k+1}{2} - k} \leq (q/(q-1))^k. \] For $s \leq k$, we show that \[ q^{-\binom{k+1}{2}} \leq |\mathcal{F}| / q^{(s-1) \binom{k+1}{2} - k} \leq (q/(q-1))^k. \] Our lower bounds rely on an iterative construction that uses lifted maximum rank-distance (MRD) codes.

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arXiv2025

Bijection Between Point-Hyperplane Anti-Flags of $V(n, 2)$ and Non-Singular Points of $O^+(2n, 2)$

We give a bijection between the point-hyperplane antiflags of $V(n, 2)$ and the nonsingular points of $V(2n, \allowbreak 2)$ 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.

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arXiv2025

On the twin-width of near-regular graphs

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.

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arXiv2025

Design switching on graphs

We show that each (r, lambda)-design yields a class of switching methods that can be used to produce cospectral graphs. We use this to explain several specific switching methods such as Godsil-McKay (GM) switching and Wang-Qiu-Hu (WQH) switching.

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arXiv2025

Intersecting Families of Spanning Trees

A family $\mathcal{F}$ of spanning trees of the complete graph on $n$ vertices $K_n$ is \emph{$t$-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 $n \geq C (\log n) t$ 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.

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arXiv2025

Ratio bound (Lovász number) versus inertia bound

Matthew Kwan and Yuval Wigderson showed that for an infinite family of graphs, the Lovász number gives an upper bound of $O(n^{3/4})$ for the size of an independent set (where $n$ is the number of vertices), while the weighted inertia bound cannot do better than $Ω(n)$. Here we point out that there is an infinite family of graphs for which the Lovász number is $Ω(n^{3/4})$, while the unweighted inertia bound is $O(n^{1/2})$.

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arXiv2025

New Constructions of Distance-Biregular Graphs

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-$\bipartB$-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.

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arXiv2025

Small genus, small index critical points of the systole function

In this paper the index of a family of critical points of the systole function on Teichm\"uller space is calculated. The members of this family are interesting in that their existence implies the existence of strata in the Thurston spine for which the systoles do not determine a basis for the homology of the surface. Previously, index calculations of critical points with this pathological feature were impossible, because the only known examples were in surfaces with huge genus. A related concept is that of a ``minimal filling subset'' of the systoles at the critical point. Such minimal filling sets are studied, as they relate to the dimension of the Thurston spine near the critical point. We find an example of a minimal filling set of simple closed geodesics in genus 5 with cardinality 8, that are presumably realised as systoles. More generally, we determine the smallest and largest cardinality of a minimal filling set related to a tesselation of a hyperbolic surface by regular, right-angled $m$-gons for $m \in \{ 5, 6, 7 \}$. For this, we use integer linear programming together with a hand-tailored symmetry breaking technique.

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arXiv2024

Regular sets of lines in rank 3 polar spaces

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 $O(7,q)$.

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arXiv2024

New bounds and constructions for large partial $m$-ovoids and related structures

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.

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arXiv2024

The classification of Boolean degree $1$ functions in high-dimensional finite vector spaces

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 $n \geq n_0(k, q)$. This also implies that two-intersecting sets with respect to $k$-spaces do not exist for $n \geq n_0(k, q)$. Our main ingredient is the Ramsey theory for geometric lattices.

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arXiv2024

A common generalization of hypercube partitions and ovoids in polar spaces

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. More precisely, for $q=p^h$, $p$ prime, and some positive integer $k$, a generalized ovoid of $k$-spaces in a polar space $\mathcal{P}$ with rank $r \geq r_0(k, p)$ in a vector space $V(n,q)$ does not exist.

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arXiv2024

The strongly regular twisted $D_{5,5}(q)$ graph

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.

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