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For a prime number $p$ and a sequence of integers $a_0,dots,a_kin {0,1,dots,p}$, let $s(a_0,dots,a_k)$ be the minimum number of $(k+1)$-tuples $(x_0,dots,x_k)in A_0timesdotstimes A_k$ with $x_0=x_1+dots + x_k$, over subsets $A_0,dots,A_ksubseteqmathbb{Z}_p$ of sizes $a_0,dots,a_k$ respectively. An elegant argument of Lev (independently rediscovered by Samotij and Sudakov) shows that there exists an extremal configuration with all sets $A_i$ being intervals of appropriate length, and that the same conclusion also holds for the related problem, reposed by Bajnok, when $a_0=dots=a_k=:a$ and $A_0=dots=A_k$, provided $k$ is not equal 1 modulo $p$. By applying basic Fourier analysis, we show for Bajnoks problem that if $pge 13$ and $ain{3,dots,p-3}$ are fixed while $kequiv 1pmod p$ tends to infinity, then the extremal configuration alternates between at least two affine non-equivalent sets.
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What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turan, Rademacher solved the first non-trivial case of this problem in 1941. The problem was revived by ErdH{o}s in 1955; it is now known as the ErdH{o}s-Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from~$1$, which in this range confirms a conjecture of Lovasz and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.
We show that the number of partial triangulations of a set of $n$ points on the plane is at least the $(n-2)$-nd Catalan number. This is tight for convex $n$-gons. We also describe all the equality cases.
We explore graph theoretical properties of minimal prime graphs of finite solvable groups. In finite group theory studying the prime graph of a group has been an important topic for the past almost half century. Recently prime graphs of solvable groups have been characterized in graph theoretical terms only. This now allows the study of these graphs with methods from graph theory only. Minimal prime graphs turn out to be of particular interest, and in this paper we pursue this further by exploring, among other things, diameters, Hamiltonian cycles and the property of being self-complementary for minimal prime graphs. We also study a new, but closely related notion of minimality for prime graphs and look into counting minimal prime graphs.
Let $p > 155$ be a prime and let $G$ be a cyclic group of order $p$. Let $S$ be a minimal zero-sum sequence with elements over $G$, i.e., the sum of elements in $S$ is zero, but no proper nontrivial subsequence of $S$ has sum zero. We call $S$ is unsplittable, if there do not exist $g$ in $S$ and $x,y in G$ such that $g=x+y$ and $Sg^{-1}xy$ is also a minimal zero-sum sequence. In this paper we show that if $S$ is an unsplittable minimal zero-sum sequence of length $|S|= frac{p-1}{2}$, then $S=g^{frac{p-11}{2}}(frac{p+3}{2}g)^4(frac{p-1}{2}g)$ or $g^{frac{p-7}{2}}(frac{p+5}{2}g)^2(frac{p-3}{2}g)$. Furthermore, if $S$ is a minimal zero-sum sequence with $|S| ge frac{p-1}{2}$, then $ind(S) leq 2$.
Let $T$ be a rooted tree, and $V(T)$ its set of vertices. A subset $X$ of $V(T)$ is called an infima closed set of $T$ if for any two vertices $u,vin X$, the first common ancestor of $u$ and $v$ is also in $X$. This paper determines the trees with minimum number of infima closed sets among all rooted trees of given order, thereby answering a question of Klazar. It is shown that these trees are essentially complete binary trees, with the exception of vertices at the last levels. Moreover, an asymptotic estimate for the minimum number of infima closed sets in a tree with $n$ vertices is also provided.
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