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Classical questions in extremal graph theory concern the asymptotics of $operatorname{ex}(G, mathcal{H})$ where $mathcal{H}$ is a fixed family of graphs and $G=G_n$ is taken from a `standard increasing sequence of host graphs $(G_1, G_2, dots)$, most often $K_n$ or $K_{n,n}$. Inverting the question, we can instead ask how large $e(G)$ can be with respect to $operatorname{ex}(G,mathcal{H})$. We show that the standard sequences indeed maximize $e(G)$ for some choices of $mathcal{H}$, but not for others. Many interesting questions and previous results arise very naturally in this context, which also, unusually, gives rise to sensible extremal questions concerning multigraphs and non-uniform hypergraphs.
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We prove that if the spectral radius of a graph G of order n is larger than the spectral radius of the r-partite Turan graph of the same order, then G contains various supergraphs of the complete graph of order r+1. In particular G contains a complete r-partite graph of size log n with one edge added to the first part. These results complete a project of Erdos from 1963. We also give corresponding stability results.
Let $t$ be an integer such that $tgeq 2$. Let $K_{2,t}^{(3)}$ denote the triple system consisting of the $2t$ triples ${a,x_i,y_i}$, ${b,x_i,y_i}$ for $1 le i le t$, where the elements $a, b, x_1, x_2, ldots, x_t,$ $y_1, y_2, ldots, y_t$ are all distinct. Let $ex(n,K_{2,t}^{(3)})$ denote the maximum size of a triple system on $n$ elements that does not contain $K_{2,t}^{(3)}$. This function was studied by Mubayi and Verstraete, where the special case $t=2$ was a problem of ErdH{o}s that was studied by various authors. Mubayi and Verstraete proved that $ex(n,K_{2,t}^{(3)})<t^4binom{n}{2}$ and that for infinitely many $n$, $ex(n,K_{2,t}^{(3)})geq frac{2t-1}{3} binom{n}{2}$. These bounds together with a standard argument show that $g(t):=lim_{nto infty} ex(n,K_{2,t}^{(3)})/binom{n}{2}$ exists and that [frac{2t-1}{3}leq g(t)leq t^4.] Addressing the question of Mubayi and Verstraete on the growth rate of $g(t)$, we prove that as $t to infty$, [g(t) = Theta(t^{1+o(1)}).]
Suppose that $R$ (red) and $B$ (blue) are two graphs on the same vertex set of size $n$, and $H$ is some graph with a red-blue coloring of its edges. How large can $R$ and $B$ be if $Rcup B$ does not contain a copy of $H$? Call the largest such integer $mathrm{mex}(n, H)$. This problem was introduced by Diwan and Mubayi, who conjectured that (except for a few specific exceptions) when $H$ is a complete graph on $k+1$ vertices with any coloring of its edges $mathrm{mex}(n,H)=mathrm{ex}(n, K_{k+1})$. This conjecture generalizes Turans theorem. Diwan and Mubayi also asked for an analogue of ErdH{o}s-Stone-Simonovits theorem in this context. We prove the following asymptotic characterization of the extremal threshold in terms of the chromatic number $chi(H)$ and the textit{reduced maximum matching number} $mathcal{M}(H)$ of $H$. $$mathrm{mex}(n, H)=left(1- frac{1}{2(chi(H)-1)} - Omegaleft(frac{mathcal{M}(H)}{chi(H)^2}right)right)frac{n^2}{2}.$$ $mathcal{M}(H)$ is, among the set of proper $chi(H)$-colorings of $H$, the largest set of disjoint pairs of color classes where each pair is connected by edges of just a single color. The result is also proved for more than $2$ colors and is tight up to the implied constant factor. We also study $mathrm{mex}(n, H)$ when $H$ is a cycle with a red-blue coloring of its edges, and we show that $mathrm{mex}(n, H)lesssim frac{1}{2}binom{n}{2}$, which is tight.
The Turan number of a graph $H$, denoted by $ex(n,H)$, is the maximum number of edges in any graph on $n$ vertices which does not contain $H$ as a subgraph. Let $P_{k}$ denote the path on $k$ vertices and let $mP_{k}$ denote $m$ disjoint copies of $P_{k}$. Bushaw and Kettle [Tur{a}n numbers of multiple paths and equibipartite forests, Combin. Probab. Comput. 20(2011) 837--853] determined the exact value of $ex(n,kP_ell)$ for large values of $n$. Yuan and Zhang [The Tur{a}n number of disjoint copies of paths, Discrete Math. 340(2)(2017) 132--139] completely determined the value of $ex(n,kP_3)$ for all $n$, and also determined $ex(n,F_m)$, where $F_m$ is the disjoint union of $m$ paths containing at most one odd path. They also determined the exact value of $ex(n,P_3cup P_{2ell+1})$ for $ngeq 2ell+4$. Recently, Bielak and Kieliszek [The Tur{a}n number of the graph $2P_5$, Discuss. Math. Graph Theory 36(2016) 683--694], Yuan and Zhang [Tur{a}n numbers for disjoint paths, arXiv: 1611.00981v1] independently determined the exact value of $ex(n,2P_5)$. In this paper, we show that $ex(n,2P_{7})=max{[n,14,7],5n-14}$ for all $n ge 14$, where $[n,14,7]=(5n+91+r(r-6))/2$, $n-13equiv r,(text{mod }6)$ and $0leq r< 6$.
For given graphs $G$ and $F$, the Turan number $ex(G,F)$ is defined to be the maximum number of edges in an $F$-free subgraph of $G$. Foucaud, Krivelevich and Perarnau and later independently Briggs and Cox introduced a dual version of this problem wherein for a given number $k$, one maximizes the number of edges in a host graph $G$ for which $ex(G,H) < k$. Addressing a problem of Briggs and Cox, we determine the asymptotic value of the inverse Turan number of the paths of length $4$ and $5$ and provide an improved lower bound for all paths of even length. Moreover, we obtain bounds on the inverse Turan number of even cycles giving improved bounds on the leading coefficient in the case of $C_4$. Finally, we give multiple conjectures concerning the asymptotic value of the inverse Turan number of $C_4$ and $P_{ell}$, suggesting that in the latter problem the asymptotic behavior depends heavily on the parity of $ell$.
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