It is not that simple.
There can be that intuition, but what you'd expect in vigorously competitive markets is for prices to move in parallel with one another. If I lower my price and want to gain market share from others, they're going to react. The competition will react to that cut in price to match my cut in price so they don't lose market share to one another.
What competition does is push prices closer to cost. One thing that could happen is costs could all vary at the same time, and if they all vary in similar ways, competitive markets will result in similar price movements. The observation of parallel behaviour is not sufficient to conclude that there's been price-fixing.
The other thing that gets very complicated and hard to deal with is that business people might recognize that there may be less than perfect competition. They might recognize that they don't want to be too aggressive on price. They could afford to be a bit aggressive on price, but they're worried there will be a response from another firm in the market and they'll match the cut in price. Then neither of them is going to be ahead. There's what's sometimes referred to as “conscious parallelism”. It's not collusion, it's not price fixing and they're not getting together. They're all just acting in their own economic interest, but it's resulting in less than perfect competition.
There are different possibilities that exist that make it difficult to observe parallel behaviour and infer any kind of price-fixing arrangement.