Strong forms of stability from flag algebra calculations

Abstract

Given a hereditary family G of admissible graphs and a function λ(G) that linearly depends on the statistics of order-κ subgraphs in a graph G, we consider the extremal problem of determining λ(n, G), the maximum of λ(G) over all admissible graphs G of order n. We call the problem perfectly B-stable for a graph B if there is a constant C such that every admissible graph G of order n C can be made into a blow-up of B by changing at most C(λ(n, G)−λ(G)) n 2 adjacencies. As special cases, this property describes all almost extremal graphs of order n within o(n 2 ) edges and shows that every extremal graph of order n n 0 is a blow-up of B.We develop general methods for establishing stability-type results from flag algebra computations and apply them to concrete examples. In fact, one of our sufficient conditions for perfect stability is stated in a way that allows automatic verification by a computer. This gives a unifying way to obtain computer-assisted proofs of many new results.

Stability results for graphs with a critical edge

“…Related results can be found in papers by Norin and Yepremyan [13,17], and Pikhurko, Sliacan and Tyros [14]. Finally, we note that results from this paper are applied in a joint paper with Natasha Morrison [11].…”

Solving Turán's Tetrahedron Problem for the $\ell_2$-Norm

“…The main power comes from the possibility of formulating a problem as a semidefinite program and using a computer to solve it. The method can be applied in various settings such as graphs [28,44], hypergraphs [3,19], oriented graphs [29,37], edge-coloured graphs [5,12], permutations [6,55], discrete geometry [7,36], or phylogenetic trees [1]. For a detailed explanation of the flag algebra method in the setting of 3-uniform hypergraphs see [22].…”

$C_5$ is almost a fractalizer

“…This type of inequality was used by Lidický and Pfender [17] when solving the Pentagon problem of Erdős for small graphs. The flag algebra method has been developed by Razborov [22], and has seen numerous applications such as [1,7,8,12,13,15,20]. We assume the reader is familiar with the method and describe only a brief outline of the calculation rather than developing the entire theory and terminology.…”

“…We assume the reader is familiar with the method and describe only a brief outline of the calculation rather than developing the entire theory and terminology. A description of the method when applied to graphs is available from several sources [3,20]. The calculation is computer assisted, and the program we used can be downloaded from the arXiv version of this paper or https://lidicky.name/pub/c5frac.…”