Hypergraphs Do Jump

Abstract

We say that α ∈ [0, 1) is a jump for an integer r 2 if there exists c(α) > 0 such that for all > 0 and all t 1, any r-graph with n n 0 (α, , t) vertices and density at least α + contains a subgraph on t vertices of density at least α + c.The Erdős-Stone-Simonovits theorem [4,5] implies that for r = 2, every α ∈ [0, 1) is a jump. Erdős [3] showed that for all r 3, every α ∈ [0, r!/r r ) is a jump. Moreover he made his famous 'jumping constant conjecture', that for all r 3, every α ∈ [0, 1) is a jump. Frankl and Rödl [7] disproved this conjecture by giving a sequence of values of non-jumps for all r 3.We use Razborov's flag algebra method [9] to show that jumps exist for r = 3 in the interval [2/9, 1). These are the first examples of jumps for any r 3 in the interval [r!/r r , 1). To be precise, we show that for r = 3 every α ∈ [0.2299, 0.2316) is a jump.We also give an improved upper bound for the Turán density of K − 4 = {123, 124, 134}: π(K − 4 ) 0.2871. This in turn implies that for r = 3 every α ∈ [0.2871, 8/27) is a jump.

Flag Algebras: An Interim Report

“…This activity started with the Mubayi challenge when all exact results presented to the author by Dhruv Mubayi found their new flagalgebraic proofs in [Raz10, Section 6.2] of varying and surprisingly unpredictable computational difficulty. [Raz10] also gave a few non-exact results, of which we would like to mention here only π(G 3 ) ≤ 0.2978 later improved by Baber and Talbot [BT11] to π(G 3 ) ≤ 0.2871 which is already quite close to the conjectured value 2/7 (see (14)). …”

“…[Raz10] that was verified in [BT11] and later in [FRV13] using the flagmatic software (we will discuss the latter in Section 4.1). The scale of this improvement reflects a general phenomenon: let me cautiously suggest that I…”

Progress on poset-free families of subsets

“…The resulting semidefinite program then requires considerable computer time to produce good bounds. This method has proven successful in Turán theory [3,45]. Theorem 3.2.…”

Infinite‐dimensional finitely forcible graphon

“…This line of research was initiated by the theory of limits of dense graphs [7][8][9]32], followed by limits of sparse graphs [5,15], permutations [24,25], partial orders [27] and others. Analytic methods applied to such limit objects led to results in many areas of mathematics and computer science, in particular in extremal combinatorics [1][2][3][4]19, 21-23, 28, 29, 37-41] and property testing [26,34].…”