arXiv subjects
Publications and source records attributed to Gabor Somlai.
Rédei and Megyesi proved that the number of directions determined by a $p$ element subset of $\mathbb{F}_p^2$ is either $1$ or at least $\frac{p+3}{2}$. The same result was independently obtained by Dress, Klin and Muzychuk. We give a new and short proof of this result using a Lemma proved by Kiss and the author. The new proof further on a result on polynomials over finite fields.
Given a triangle $Δ$, we study the problem of determining the smallest enclosing and largest embedded isosceles triangles of $Δ$ with respect to area and perimeter. This problem was initially posed by Nandakumar and was first studied by Kiss, Pach, and Somlai, who showed that if $Δ'$ is the smallest area isosceles triangle containing $Δ$, then $Δ'$ and $Δ$ share a side and an angle. In the present paper, we prove that for any triangle $Δ$, every maximum area isosceles triangle embedded in $Δ$ and every maximum perimeter isosceles triangle embedded in $Δ$ shares a side and an angle with $Δ$. Somewhat surprisingly, the case of minimum perimeter enclosing triangles is different: there are infinite families of triangles $Δ$ whose minimum perimeter isosceles containers do not share a side and an angle with $Δ$.
For every infinite sequence of simple groups of Lie type of growing rank we exhibit connected Cayley graphs of degree at most 10 such that the isoperimetric number of these graphs converges to 0. This proves that these graphs do not form a family of expanders.
For every prime $p>3$ we prove that $Q \times \mathbb{Z}_p$ and $\mathbb{Z}_2^3 \times \mathbb{Z}_p$ are DCI- groups. This result completes the description of CI-groups of order $8p$.
For every prime $p > 2$ we exhibit a Cayley graph of $\mathbb{Z}_p^{2p+3}$ which is not a CI-graph. This proves that an elementary Abelian $p$-group of rank greater than or equal to $2p+3$ is not a CI-group. The proof is elementary and uses only multivariate polynomials and basic tools of linear algebra. Moreover, we apply our technique to give a uniform explanation for the recent works concerning the bound.