arXiv subjects

Barnabás Janzer

Publications and source records attributed to Barnabás Janzer.

22 records · Page 2Linked to original sources

arXiv2020

Generalizations of the Ruzsa-Szemerédi and rainbow Turán problems for cliques

Considering a natural generalization of the Ruzsa-Szemerédi problem, we prove that for any fixed positive integers $r,s$ with $r<s$, there are graphs on $n$ vertices containing $n^{r}e^{-O(\sqrt{\log{n}})}=n^{r-o(1)}$ copies of $K_s$ such that any $K_r$ is contained in at most one $K_s$. We also give bounds for the generalized rainbow Turán problem $\operatorname{ex}(n, H,$rainbow-$F)$ when $F$ is complete. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer, showing that there are properly edge-coloured graphs on $n$ vertices with $n^{r-1-o(1)}$ copies of $K_r$ such that no $K_r$ is rainbow.

math.CO↗

arXiv2019

Projections of antichains

A subset $A$ of $\mathbb{Z}^n$ is called a weak antichain if it does not contain two elements $x$ and $y$ satisfying $x_i<y_i$ for all $i$. Engel, Mitsis, Pelekis and Reiher showed that for any weak antichain $A$, the sum of the sizes of its $(n-1)$-dimensional projections must be at least as large as its size $|A|$. They asked what the smallest possible value of the gap between these two quantities is in terms of $|A|$. We answer this question by giving an explicit weak antichain attaining this minimum for each possible value of $|A|$. In particular, we show that sets of the form $A_N=\{x\in\mathbb{Z}^n: 0\leq x_j\leq N-1$ for all $j$ and $x_i=0$ for some $i\}$ minimise the gap among weak antichains of size $|A_N|$.

math.CO↗

arXiv2019

A note on antichains in the continuous cube

It is well-known that an antichain in the poset $[0,1]^n$ must have measure zero. Engel, Mitsis, Pelekis and Reiher showed that in fact it must have $(n-1)$-dimensional Hausdorff measure at most $n$, and they conjectured that this bound can be attained. In this note we show that, for every $n$, such an antichain does indeed exist.

math.CO↗

arXiv2017

A New Upper Bound for Cancellative Pairs

A pair $(\mathcal{A},\mathcal{B})$ of families of subsets of an $n$-element set is called cancellative if whenever $A,A'\in\mathcal{A}$ and $B\in\mathcal{B}$ satisfy $A\cup B=A'\cup B$, then $A=A'$, and whenever $A\in\mathcal{A}$ and $B,B'\in\mathcal{B}$ satisfy $A\cup B=A\cup B'$, then $B=B'$. It is known that there exist cancellative pairs with $|\mathcal{A}||\mathcal{B}|$ about $2.25^n$, whereas the best known upper bound on this quantity is $2.3264^n$. In this paper we improve this upper bound to $2.2682^n$. Our result also improves the best known upper bound for Simonyi's sandglass conjecture for set systems.

math.CO↗