Fifty years ago Erdős asked to determine the minimum number of k-cliques in a graph on n vertices with independence number less than l. He conjectured that this minimum is achieved by the disjoint union of l − 1 complete graphs of size n l−1 . This conjecture was disproved by Nikiforov who showed that the balanced blow-up of a 5-cycle has fewer 4-cliques than the union of 2 complete graphs of size n 2 . In this paper we solve Erdős' problem for (k, l) = (3, 4) and (k, l) = (4, 3). Using stability arguments we also characterize the precise structure of extremal examples, confirming Erdős' conjecture for (k, l) = (3, 4) and showing that a blow-up of a 5-cycle gives the minimum for (k, l) = (4, 3).
We present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker-Breaker games. In particular, we show that for b = o ( √ n), Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a (1 : b) game on E(K n ). As another application, we show that for b = Θ (n/ ln n), playing a (1 : b) game on E(K n ), Maker can build a graph which contains copies of all spanning trees having maximum degree ∆ = O(1) with a bare path of linear length (a bare path in a tree T is a path with all interior vertices of degree exactly two in T ).
A conjecture of Bandelt and Dress states that the maximum quartet distance between any two phylogenetic trees on n leaves is at most (. Using the machinery of flag algebras we improve the currently known bounds regarding this conjecture, in particular we show that the maximum is at most (0.69 + o(1)) n 4 . We also give further evidence that the conjecture is true by proving that the maximum distance between caterpillar trees is at most (
For a graph G, let pi(G),i=0,...,3 be the probability that three distinct random vertices span exactly i edges. We call (p0(G),...,p3(G)) the 3‐local profile
of G. We investigate the set scriptS3⊂double-struckR4 of all vectors (p0,...,p3) that are arbitrarily close to the 3‐local profiles of arbitrarily large graphs. We give a full description of the projection of S3 to the (p0,p3) plane. The upper envelope of this planar domain is obtained from cliques on a fraction of the vertex set and complements of such graphs. The lower envelope is Goodman's inequality p0+p3≥14. We also give a full description of the triangle‐free case, i.e. the intersection of S3 with the hyperplane p3=0. This planar domain is characterized by an SDP constraint that is derived from Razborov's flag algebra theory.
Let r, s ≥ 2 be integers. Suppose that the number of blue r-cliques in a red/blue coloring of the edges of the complete graph Kn is known and fixed. What is the largest possible number of red s-cliques under this assumption? The well known Kruskal-Katona theorem answers this question for r = 2 or s = 2. Using the shifting technique from extremal set theory together with some analytical arguments, we resolve this problem in general and prove that in the extremal coloring either the blue edges or the red edges form a clique.
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