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The permutohedron $P_n$ of order $n$ is a polytope embedded in $\mathbb{R}^n$ whose vertex coordinates are permutations of the first $n$ natural numbers. It is obvious that $P_n$ lies on the hyperplane $H_n$ consisting of points whose coordinates sum up to $n(n+1)/2$. We prove that if the vertices of $P_n$ are contained in the union of $m$ affine hyperplanes different from $H_n$, then $m\geq n$ when $n \geq 3$ is odd, and $m \geq n-1$ when $n \geq 4$ is even. This confirms a recent conjecture of Hegedüs and Károlyi. Our proof gives an algebraic criterion for a non-standard permutohedron generated by $n$ distinct real numbers to require at least $n$ non-trivial hyperplanes to cover its vertices.

The generalized Turán number $ex(n,K_s,H)$ is the maximum number of complete graph $K_s$ in an $H$-free graph on $n$ vertices. Let $F_k$ be the friendship graph consisting of $k$ triangles. Erdős and Sós (1976) determined the value of $ex(n,K_3,F_2)$. Alon and Shikhelman (2016) proved that $ex(n,K_3, F_k)\le (9k-15)(k+1)n.$ In this paper, by using a method developed by Chung and Frankl in hypergraph theory, we determine the exact value of $ex(n,K_3,F_k)$ and the extremal graph for any $F_k$ when $n\ge 4k^3$.

We prove a fractional Helly theorem for $k$-flats intersecting fat convex sets. A family $\mathcal{F}$ of sets is said to be $\rho$-fat if every set in the family contains a ball and is contained in a ball such that the ratio of the radii of these balls is bounded by $\rho$. We prove that for every dimension $d$ and positive reals $\rho$ and $\alpha$ there exists a positive $\beta=\beta(d,\rho, \alpha)$ such that if $\mathcal{F}$ is a finite family of $\rho$-fat convex sets in $\mathbb{R}^d$ and an $\alpha$-fraction of the $(k+2)$-size subfamilies from $\mathcal{F}$ can be hit by a $k$-flat, then there is a $k$-flat that intersects at least a $\beta$-fraction of the sets of $\mathcal{F}$. We prove spherical and colorful variants of the above results and prove a $(p,k+2)$-theorem for $k$-flats intersecting balls.

The recently introduced \emph{Degree Preserving Growth} model (Nature Physics, \DOI{https://doi.org/10.1038/s41567-021-01417-7}) uses matchings to insert new vertices of prescribed degrees into the current graph of an ever-growing graph sequence. The process depends both on the size of the largest available matchings, which is our focus here, as well as on the actual choice of the matching. First we show that the question whether a graphic degree sequence, extended with a new degree $2\delta$ remains graphic is closely related to the available matchings in the realizations of the sequence. Namely we prove that the extension problem is equivalent to the existence of a realization of the original degree sequence with a matching of size $\delta$. Second we present lower bounds for the \emph{forcible matching number} of degree sequences. This number is the size of the maximum matchings in any realization of the degree sequence. We then study bounds on the size of maximal matchings in \emph{some} realizations of the sequence, known as the \emph{potential matching number}. We also estimate the minimum size of both the maximal and the maximum matchings, as determined by the degree sequence, independently of graphical realizations. Along this line we answer a question raised by Biedl, Demaine \emph{et al.} (\DOI{https://doi.org/10.1016/j.disc.2004.05.003}).

We study the query complexity on slices of Boolean functions. Among other results we show that there exists a Boolean function for which we need to query all but 7 input bits to compute its value, even if we know beforehand that the number of 0's and 1's in the input are the same, i.e., when our input is from the middle slice. This answers a question of Byramji. Our proof is non-constructive, but we also propose a concrete candidate function that might have the above property. Our results are related to certain natural discrepancy type questions that, somewhat surprisingly, have not been studied before.

The systematic study of Turán-type extremal problems for edge-ordered graphs was initiated by Gerbner et al. in 2020. Here we characterize connected edge-ordered graphs with linear extremal functions and show that the extremal function of other connected edge-ordered graphs is $\Omega(n\log n)$. This characterization and dichotomy are similar in spirit to results of Füredi et al. (2020) about vertex-ordered and convex geometric graphs. We also extend the study of extremal function of short edge-ordered paths by Gerbner et al. to some longer paths.

Fix a $k$-chromatic graph $F$. In this paper we consider the question to determine for which graphs $H$ does the Turán graph $T_{k-1}(n)$ have the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large enough). We say that such a graph $H$ is $F$-Turán-good. In addition to some general results, we give (among others) the following concrete results: (i) For every complete multipartite graph $H$, there is $k$ large enough such that $H$ is $K_k$-Turán-good. (ii) The path $P_3$ is $F$-Turán-good for $F$ with $\chi(F) \geq 4$. (iii) The path $P_4$ and cycle $C_4$ are $C_5$-Turán-good. (iv) The cycle $C_4$ is $F_2$-Turán-good where $F_2$ is the graph of two triangles sharing exactly one vertex.