The first application of Szemerédi's powerful regularity method was the following celebrated Rams... more The first application of Szemerédi's powerful regularity method was the following celebrated Ramsey-Turán result proved by Szemerédi in 1972: any K 4-free graph on n vertices with independence number o(n) has at most (1 8 +o(1))n 2 edges. Four years later, Bollobás and Erdős gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K 4-free graph on n vertices with independence number o(n) and (1 8 − o(1))n 2 edges. Starting with Bollobás and Erdős in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about n 2 /8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.
A graph G is Ramsey for H if every two-colouring of the edges of G contains a monochromatic copy ... more A graph G is Ramsey for H if every two-colouring of the edges of G contains a monochromatic copy of H. Two graphs H and H are Ramsey-equivalent if every graph G is Ramsey for H if and only if it is Ramsey for H. In this paper, we study the problem of determining which graphs are Ramsey-equivalent to the complete graph K k. A famous theorem of Nešetřil and Rödl implies that any graph H which is Ramsey-equivalent to K k must contain K k. We prove that the only connected graph which is Ramsey-equivalent to K k is itself. This gives a negative answer to the question of Szabó, Zumstein, and Zürcher on whether K k is Ramsey-equivalent to K k • K 2 , the graph on k + 1 vertices consisting of K k with a pendent edge. In fact, we prove a stronger result. A graph G is Ramsey minimal for a graph H if it is Ramsey for H but no proper subgraph of G is Ramsey for H. Let s(H) be the smallest minimum degree over all Ramsey minimal graphs for H. The study of s(H) was introduced by Burr, Erdős, and Lovász, where they show that s(K k) = (k −1) 2. We prove that s(K k • K 2) = k − 1, and hence K k and K k • K 2 are not Ramsey-equivalent. We also address the question of which non-connected graphs are Ramsey-equivalent to K k. Let f (k, t) be the maximum f such that the graph H = K k + f K t , consisting of K k and f disjoint copies of K t , is Ramsey-equivalent to K k. Szabó, Zumstein, and Zürcher gave a lower bound on f (k, t). We prove an upper bound on f (k, t) which is roughly within a factor 2 of the lower bound.
Ramsey's theorem, in the version of Erdős and Szekeres, states that every 2-coloring of the edges... more Ramsey's theorem, in the version of Erdős and Szekeres, states that every 2-coloring of the edges of the complete graph on {1, 2,. .. , n} contains a monochromatic clique of order 1 2 log n. In this paper, we consider two well-studied extensions of Ramsey's theorem. Improving a result of Rödl, we show that there is a constant c > 0 such that every 2-coloring of the edges of the complete graph on {2, 3, ..., n} contains a monochromatic clique S for which the sum of 1/ log i over all vertices i ∈ S is at least c log log log n. This is tight up to the constant factor c and answers a question of Erdős from 1981. Motivated by a problem in model theory, Väänänen asked whether for every k there is an n such that the following holds. For every permutation π of 1,. .. , k − 1, every 2-coloring of the edges of the complete graph on {1, 2,. .. , n} contains a monochromatic clique a 1 <. .. < a k with a π(1)+1 − a π(1) > a π(2)+1 − a π(2) >. .. > a π(k−1)+1 − a π(k−1). That is, not only do we want a monochromatic clique, but the differences between consecutive vertices must satisfy a prescribed order. Alon and, independently, Erdős, Hajnal and Pach answered this question affirmatively. Alon further conjectured that the true growth rate should be exponential in k. We make progress towards this conjecture, obtaining an upper bound on n which is exponential in a power of k. This improves a result of Shelah, who showed that n is at most double-exponential in k.
The first application of Szemerédi's powerful regularity method was the following celebrated Rams... more The first application of Szemerédi's powerful regularity method was the following celebrated Ramsey-Turán result proved by Szemerédi in 1972: any K 4-free graph on n vertices with independence number o(n) has at most (1 8 +o(1))n 2 edges. Four years later, Bollobás and Erdős gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K 4-free graph on n vertices with independence number o(n) and (1 8 − o(1))n 2 edges. Starting with Bollobás and Erdős in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about n 2 /8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.
A graph G is Ramsey for H if every two-colouring of the edges of G contains a monochromatic copy ... more A graph G is Ramsey for H if every two-colouring of the edges of G contains a monochromatic copy of H. Two graphs H and H are Ramsey-equivalent if every graph G is Ramsey for H if and only if it is Ramsey for H. In this paper, we study the problem of determining which graphs are Ramsey-equivalent to the complete graph K k. A famous theorem of Nešetřil and Rödl implies that any graph H which is Ramsey-equivalent to K k must contain K k. We prove that the only connected graph which is Ramsey-equivalent to K k is itself. This gives a negative answer to the question of Szabó, Zumstein, and Zürcher on whether K k is Ramsey-equivalent to K k • K 2 , the graph on k + 1 vertices consisting of K k with a pendent edge. In fact, we prove a stronger result. A graph G is Ramsey minimal for a graph H if it is Ramsey for H but no proper subgraph of G is Ramsey for H. Let s(H) be the smallest minimum degree over all Ramsey minimal graphs for H. The study of s(H) was introduced by Burr, Erdős, and Lovász, where they show that s(K k) = (k −1) 2. We prove that s(K k • K 2) = k − 1, and hence K k and K k • K 2 are not Ramsey-equivalent. We also address the question of which non-connected graphs are Ramsey-equivalent to K k. Let f (k, t) be the maximum f such that the graph H = K k + f K t , consisting of K k and f disjoint copies of K t , is Ramsey-equivalent to K k. Szabó, Zumstein, and Zürcher gave a lower bound on f (k, t). We prove an upper bound on f (k, t) which is roughly within a factor 2 of the lower bound.
Ramsey's theorem, in the version of Erdős and Szekeres, states that every 2-coloring of the edges... more Ramsey's theorem, in the version of Erdős and Szekeres, states that every 2-coloring of the edges of the complete graph on {1, 2,. .. , n} contains a monochromatic clique of order 1 2 log n. In this paper, we consider two well-studied extensions of Ramsey's theorem. Improving a result of Rödl, we show that there is a constant c > 0 such that every 2-coloring of the edges of the complete graph on {2, 3, ..., n} contains a monochromatic clique S for which the sum of 1/ log i over all vertices i ∈ S is at least c log log log n. This is tight up to the constant factor c and answers a question of Erdős from 1981. Motivated by a problem in model theory, Väänänen asked whether for every k there is an n such that the following holds. For every permutation π of 1,. .. , k − 1, every 2-coloring of the edges of the complete graph on {1, 2,. .. , n} contains a monochromatic clique a 1 <. .. < a k with a π(1)+1 − a π(1) > a π(2)+1 − a π(2) >. .. > a π(k−1)+1 − a π(k−1). That is, not only do we want a monochromatic clique, but the differences between consecutive vertices must satisfy a prescribed order. Alon and, independently, Erdős, Hajnal and Pach answered this question affirmatively. Alon further conjectured that the true growth rate should be exponential in k. We make progress towards this conjecture, obtaining an upper bound on n which is exponential in a power of k. This improves a result of Shelah, who showed that n is at most double-exponential in k.
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Papers by Jacob Fox