Short proofs of some extremal results

We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.Comment: 19 page

Short proofs of some extremal results III

We prove a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are reasonably short

Short proofs of some extremal results III

We prove a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are reasonably short

On the universal method to solve extremal problems

Some applications of the theory of extremal problems to mathematics and economics are made more accessible to non-experts.1.The following fundamental results are known to all users of mathematical techniques, such as economist, econometricians, engineers and ecologists: the fundamental theorem of algebra, the Lagrange multiplier rule, the implicit function theorem, separation theorems for convex sets, orthogonal diagonalization of symmetric matrices. However, full explanations, including rigorous proofs, are only given in relatively advanced courses for mathematicians. Here, we offer short ans easy proofs. We show that akk these results can be reduced to the task os solving a suitable extremal problem. Then we solve each of the resulting problems by a universal strategy.2. The following three practical results, each earning their discoverers the Nobel prize for Economics, are known to all economists and aonometricians: Nash bargaining, the formula of Black and Scholes for the price of options and the models of Prescott and Kydland on the value of commitment. However, the great value of such applications of the theory of extremal problems deserves to be more generally appreciated. The great impact of these results on real life examples is explained. This, rather than mathematical depth, is the correct criterion for assessing their value.

On the universal method to solve extremal problems

Some applications of the theory of extremal problems to mathematics and economics are made more accessible to non-experts. 1.The following fundamental results are known to all users of mathematical techniques, such as economist, econometricians, engineers and ecologists: the fundamental theorem of algebra, the Lagrange multiplier rule, the implicit function theorem, separation theorems for convex sets, orthogonal diagonalization of symmetric matrices. However, full explanations, including rigorous proofs, are only given in relatively advanced courses for mathematicians. Here, we offer short ans easy proofs. We show that akk these results can be reduced to the task os solving a suitable extremal problem. Then we solve each of the resulting problems by a universal strategy. 2. The following three practical results, each earning their discoverers the Nobel prize for Economics, are known to all economists and aonometricians: Nash bargaining, the formula of Black and Scholes for the price of options and the models of Prescott and Kydland on the value of commitment. However, the great value of such applications of the theory of extremal problems deserves to be more generally appreciated. The great impact of these results on real life examples is explained. This, rather than mathematical depth, is the correct criterion for assessing their value

Extremal problems for the p-spectral radius of graphs

The $p$-spectral radius of a graph $G\$of order $n$ is defined for any real number $p\geq1$ as $$\lambda^{\left( p\right) }\left( G\right) =\max\left\{ 2\sum_{\{i,j\}\in E\left( G\right) \ }x_{i}x_{j}:x_{1},\ldots,x_{n}\in\mathbb{R}\text{ and }\left\vert x_{1}\right\vert ^{p}+\cdots+\left\vert x_{n}\right\vert ^{p}=1\right\} .$$ The most remarkable feature of $\lambda^{\left( p\right) }$ is that it seamlessly joins several other graph parameters, e.g., $\lambda^{\left( 1\right) }$ is the Lagrangian, $\lambda^{\left( 2\right) }$ is the spectral radius and $\lambda^{\left( \infty\right) }/2$ is the number of edges. This paper presents solutions to some extremal problems about $\lambda^{\left( p\right) }$, which are common generalizations of corresponding edge and spectral extremal problems. Let $T_{r}\left( n\right)$ be the $r$-partite Tur\'{a}n graph of order $n.$ Two of the main results in the paper are: (I) Let $r\geq2$ and $p>1.$ If $G$ is a $K_{r+1}$-free graph of order $n,$ then $$\lambda^{\left( p\right) }\left( G\right) <\lambda^{\left( p\right) }\left( T_{r}\left( n\right) \right) ,$$ unless $G=T_{r}\left( n\right) .$ (II) Let $r\geq2$ and $p>1.$ If $G\$is a graph of order $n,$ with $$\lambda^{\left( p\right) }\left( G\right) >\lambda^{\left( p\right) }\left( T_{r}\left( n\right) \right) ,$$ then $G$ has an edge contained in at least $cn^{r-1}$ cliques of order $r+1,$ where $c$ is a positive number depending only on $p$ and $r.$Comment: 21 pages. Some minor corrections in v

Maxima of the Q-index: forbidden 4-cycle and 5-cycle

This paper gives tight upper bounds on the largest eigenvalue q(G) of the signless Laplacian of graphs with no 4-cycle and no 5-cycle. If n is odd, let F_{n} be the friendship graph of order n; if n is even, let F_{n} be F_{n-1} with an edge hanged to its center. It is shown that if G is a graph of order n, with no 4-cycle, then q(G)<q(F_{n}), unless G=F_{n}. Let S_{n,k} be the join of a complete graph of order k and an independent set of order n-k. It is shown that if G is a graph of order n, with no 5-cycle, then q(G)<q(S_{n,2}), unless G=S_{n,k}. It is shown that these results are significant in spectral extremal graph problems. Two conjectures are formulated for the maximum q(G) of graphs with forbidden cycles.Comment: 12 page