Turán's famous tetrahedron problem is to compute the Turán density of the tetrahedron K43$K_4^3$. This is equivalent to determining the maximum ℓ1$\ell _1$‐norm of the codegree vector of a K43$K_4^3$‐free n$n$‐vertex 3‐uniform hypergraph. We introduce a new way for measuring extremality of hypergraphs and determine asymptotically the extremal function of the tetrahedron in our notion. The codegree squared sum, co2(G)$\mbox{co}_2(G)$, of a 3‐uniform hypergraph G$G$ is the sum of codegrees squared dfalse(x,yfalse)2$d(x,y)^2$ over all pairs of vertices xy$xy$, or in other words, the square of the ℓ2$\ell _2$‐norm of the codegree vector of the pairs of vertices. We define exco2(n,H)$\mbox{exco}_2(n,H)$ to be the maximum co2(G)$\mbox{co}_2(G)$ over all H$H$‐free n$n$‐vertex 3‐uniform hypergraphs G$G$. We use flag algebra computations to determine asymptotically the codegree squared extremal number for K43$K_4^3$ and K53$K_5^3$ and additionally prove stability results. In particular, we prove that the extremal K43$K_4^3$‐free hypergraphs in ℓ2$\ell _2$‐norm have approximately the same structure as one of the conjectured extremal hypergraphs for Turán's conjecture. Further, we prove several general properties about exco2(n,H)$\mbox{exco}_2(n,H)$ including the existence of a scaled limit, blow‐up invariance and a supersaturation result.
An edge coloring of a complete graph with a set of colors C is called completely balanced if any vertex is incident to the same number of edges of each color from C. Erdős and Tuza asked in 1993 whether for any graph F on ℓ edges and any completely balanced coloring of any sufficiently large complete graph using ℓ colors contains a rainbow copy of F . We answer this and a related question in the negative for an infinite family of graphs F by giving an explicit construction of a respective completely balanced coloring.
The Erdős–Simonovits stability theorem states that for all ε > 0 there exists α > 0 such that if G is a Kr+1-free graph on n vertices with e(G) > ex(n, Kr+1)– α n2, then one can remove εn2 edges from G to obtain an r-partite graph. Füredi gave a short proof that one can choose α = ε. We give a bound for the relationship of α and ε which is asymptotically sharp as ε → 0.
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