arXiv subjects

Csongor Beke

Publications and source records attributed to Csongor Beke.

6 recordsLinked to original sources

arXiv2026

A characterization of idempotent Schur multipliers

We prove that every idempotent Schur multiplier is a finite signed sum of contractive idempotent Schur multipliers. This was conjectured by Katavolos and Paulsen in 2003 and previously known only for translation-invariant Schur multipliers, by the Cohen-Host idempotent theorem. Concretely, we show that any boolean matrix $A$ with Schur multiplier norm at most~$\gamma$ (or equivalently $\lVert A\rVert_{\gamma_2} \le \gamma$) can be written as \[ A=\sum_{i=1}^{L}\sigma_i B_i,\] where $L\leq 2^{C\gamma^6}$ for an absolute constant $C$, $\sigma_i\in\{-1,1\}$ are signs, and each $B_i$ is a contractive idempotent Schur multiplier, that is, a boolean matrix whose $1$-entries form a union of all-one rectangular blocks, with no two blocks sharing a row or a column.

math.CA↗

arXiv2025

The multicolour size Ramsey number of a path

In this paper, we determine the $r$-colour size Ramsey number of the path $P_k$, up to constants. In particular, for every fixed $r \geq 2$ and $k \geq 100\log r$, we have \[ \widehat{R}_r(P_k)=\Theta((r^2 \log r) \, k).\] Perhaps surprisingly, we do this by improving the lower bound on $\widehat{R}_r(P_k)$.

math.CO↗

arXiv2024

Permutation Tutte polynomial

The classical Tutte polynomial is a two-variate polynomial $T_G(x,y)$ associated to graphs or more generally, matroids. In this paper, we introduce a polynomial $\widetilde{T}_H(x,y)$ associated to a bipartite graph $H$ that we call the permutation Tutte polynomial of the graph $H$. It turns out that $T_G(x,y)$ and $\widetilde{T}_H(x,y)$ share many properties, and the permutation Tutte polynomial serves as a tool to study the classical Tutte polynomial. We discuss the analogs of Brylawsi's identities and Conde--Merino--Welsh type inequalities. In particular, we will show that if $H$ does not contain isolated vertices, then $$\widetilde{T}_H(3,0)\widetilde{T}_H(0,3)\geq \widetilde{T}_H(1,1)^2,$$ which gives a short proof to the analogous result of Jackson: $$T_G(3,0)T_G(0,3)\geq T_G(1,1)^2$$ for graphs without loops and bridges. We also improve on the constant $3$ in this statement by showing that one can replace it with $2.9243$.

math.CO↗

arXiv2024

The Merino--Welsh conjecture is false for matroids

The matroidal version of the Merino--Welsh conjecture states that the Tutte polynomial $T_M(x,y)$ of any matroid $M$ without loops and coloops satisfies that $$\max(T_M(2,0),T_M(0,2))\geq T_M(1,1).$$ Equivalently, if the Merino--Welsh conjecture is true for all matroids without loops and coloops, then the following inequalities are also satisfied for all matroids without loops and coloops: $$T_M(2,0)+T_M(0,2)\geq 2T_M(1,1),$$ and $$T_M(2,0)T_M(0,2)\geq T_M(1,1)^2.$$ We show a counter-example for these inequalities.

math.CO↗

arXiv2023

On the generalized Turán problem for odd cycles

In 1984, Erdős conjectured that the number of pentagons in any triangle-free graph on $n$ vertices is at most $(n/5)^5$, which is sharp by the balanced blow-up of a pentagon. This was proved by Grzesik, and independently by Hatami, Hladký, Král', Norine and Razborov. As an extension of this result for longer cycles, we prove that for each odd $k\geq 7$, the balanced blow-up of $C_k$ (uniquely) maximises the number of $k$-cycles among $C_{k-2}$-free graphs on $n$ vertices, as long as $n$ is sufficiently large. We also show that this is no longer true if $n$ is not assumed to be sufficiently large. Our result strengthens results of Grzesik and Kielak who proved that for each odd $k\geq 7$, the balanced blow-up of $C_k$ maximises the number of $k$-cycles among graphs with a given number of vertices and no odd cycles of length less than $k$. We further show that if $k$ and $\ell$ are odd and $k$ is sufficiently large compared to $\ell$, then the balanced blow-up of $C_{\ell+2}$ does not asymptotically maximise the number of $k$-cycles among $C_{\ell}$-free graphs on $n$ vertices. This disproves a conjecture of Grzesik and Kielak.

math.CO↗

arXiv2022

Short proof of a theorem of Brylawski on the coefficients of the Tutte polynomial

In this short note we show that a system $M=(E,r)$ with a ground set $E$ of size $m$ and (rank) function $r: 2^E\to \mathbb{Z}_{\geq 0}$ satisfying $r(S)\leq \min(r(E),|S|)$ for every set $S\subseteq E$, the Tutte polynomial $$T_M(x,y):=\sum_{S\subseteq E}(x-1)^{r(E)-r(S)}(y-1)^{|S|-r(S)},$$ written as $T_M(x,y)=\sum_{i,j}t_{ij}x^iy^j$, satisfies that for any integer $h \geq 0$, we have $$\sum_{i=0}^h\sum_{j=0}^{h-i}\binom{h-i}{j}(-1)^jt_{ij}=(-1)^{m-r}\binom{h-r}{h-m},$$ where $r=r(E)$, and we use the convention that when $h<m$, the binomial coefficient $\binom{h-r}{h-m}$ is interpreted as $0$. This generalizes a theorem of Brylawski on matroid rank functions and $h<m$, and a theorem of Gordon for $h\leq m$ with the same assumptions on the rank function. The proof presented here is significantly shorter than the previous ones. We only use the fact that the Tutte polynomial $T_M(x,y)$ simplifies to $(x-1)^{r(E)}y^{|E|}$ along the hyperbola $(x-1)(y-1)=1$.

math.CO↗