The classical stability theorem of Erdős and Simonovits states that, for any fixed graph with chromatic number k + 1 ≥ 3, the following holds: every n-vertex graph that is H-free and has within o(n 2 ) of the maximal possible number of edges can be made into the k-partite Turán graph by adding and deleting o(n 2 ) edges. In this paper, we prove sharper quantitative results for graphs H with a critical edge, both for the Erdős-Simonovits Theorem (distance to the Turán graph) and for the closely related question of how close an H-free graph is to being k-partite. In many cases, these results are optimal to within a constant factor.
Turán's Theorem says that an extremal Kr+1-free graph is r-partite. The Stability Theorem of Erdős and Simonovits shows that if a Kr+1-free graph with n vertices has close to the maximal tr(n) edges, then it is close to being r-partite. In this paper we determine exactly the Kr+1-free graphs with at least m edges that are farthest from being r-partite, for any m≥tr(n)−δrn2. This extends work by Erdős, Győri and Simonovits, and proves a conjecture of Balogh, Clemen, Lavrov, Lidický and Pfender.
Mossel and Ross raised the question of when a random coloring of a graph can be reconstructed from local information, namely, the colorings (with multiplicity) of balls of given radius. In this article, we are concerned with random 2-colorings of the vertices of the n-dimensional hypercube, or equivalently random Boolean functions. In the worst case, balls of diameter Ω(n) are required to reconstruct. However, the situation for random colorings is dramatically different: we show that almost every 2-coloring can be reconstructed from the multiset of colorings of balls of radius 2. Furthermore, we show that for q ≥ n 2+𝜖 , almost every q-coloring can be reconstructed from the multiset of colorings of 1-balls.
Fix k ≥ 2 and let H be a graph with χ(H) = k + 1 containing a critical edge. We show that for sufficiently large n, the unique n-vertex H-free graph containing the maximum number of cycles is T k (n). This resolves both a question and a conjecture of Arman,
Harper's Theorem states that in a hypercube the Hamming balls have minimal vertex boundaries with respect to set size. In this paper we prove a stability-like result for Harper's Theorem: if the vertex boundary of a set is close to minimal in the hypercube, then the set must be very close to a Hamming ball around some vertex.
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